WONDERFUL VARIETIES OF TYPE D

arXiv:math/0410472v1 [math.RT] 21 Oct 2004

WONDERFUL VARIETIES OF TYPE D PAOLO BRAVI AND GUIDO PEZZINI Abstract. Let G be a connected semisimple group over C, whose simple components have type A or D. We prove that wonderful G-varieties are classified by means of combinatorial objects called spherical systems. This is a generalization of a known result of Luna for groups of type A; thanks to another result of Luna, this implies also the classification of all spherical G-varieties for the groups G we are considering. For these G we also prove the smoothness of the embedding of Demazure.

1. Introduction Let G be a connected semisimple group over C. In [L3] it is proven that if G is adjoint of type A then wonderful G-varieties are classified by means of discrete invariants coming from the theory of spherical varieties. These invariants are treated as combinatorial objects called spherical systems. In this work we extend this result to the case where G has simple components of type A or D. Wonderful varieties were studied at first in the theory of linear algebraic groups as compactifications of certain homogeneous spaces, precisely the symmetric ones (for more details, see [DP]). They also appear in the theory of spherical embeddings, as “canonical” compactifications of particular spherical homogeneous spaces. Actually all wonderful varieties can be obtained in this way ([L1]). Wonderful varieties of low rank have been classified for all G, by Ahiezer (rank 1, in the framework of compact groups, also obtained with algebraic methods by Brion) and Wasserman (rank 2). Their results have been crucial in establishing the axioms defining the spherical systems. The role of wonderful varieties within the theory of spherical varieties has been made clear in [Kn] and [L3]; their importance has grown up to the point that the classification obtained in [L3] and in the present work implies the classification of all spherical G-varieties for the groups G under consideration. We also prove for these groups a known conjecture of Brion (appeared in [B3]), already proven by Luna in [L4] for G of type A, which states that the embedding of Demazure is smooth. We are grateful to D. Luna and C. Procesi for their precious help. 2. The classification of wonderful varieties 2.1. Wonderful varieties. We recall some basic definitions and known facts about wonderful and spherical varieties. Let G be a complex semisimple linear algebraic group, B and B − two mutually opposite Borel subgroups, and T = B ∩ B − a maximal torus. 2000 Mathematics Subject Classification. 14L30 (14M17). 1

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PAOLO BRAVI AND GUIDO PEZZINI

Definition 2.1 ([DP]). An algebraic G-variety X is wonderful of rank r if: - X is smooth and complete, - G has a dense orbit in X whose complement is the union of r smooth prime divisors Di , i = 1, . . . , r, with normal crossings, - the intersection of the divisors Di is nonempty and, for all I ⊆ {1, . . . , r}, \ [ ( Di ) \ ( Di ) i∈I

i∈I /

is a G-orbit. A wonderful G-variety is always projective and spherical (see [L1]), where in general an irreducible G-variety X is said to be spherical if it is normal and it contains an open dense B-orbit. This implies that any B-proper rational function on X is uniquely determined (up to a scalar factor) by its B-weight; the lattice of such weights is called ΞX . If X is wonderful of rank r then ΞX has rank r ([L1]). Every Q-valued discrete valuation ν of C(X) defines a functional ρX (ν) on ΞX in the following way: hρX (ν), χi = ν(fχ ) where χ ∈ ΞX and fχ ∈ C(X) is B-proper with weight χ. The application ν 7→ ρX (ν) is injective if we take its restriction to the set VX of all G-invariant valuations of X. In this way VX is identified with a subset of HomZ (ΞX , Q), and it turns out to be a polyhedral convex cone. This cone can be described as the set {v ∈ HomZ (ΞX , Q) : v(χ) ≤ 0 ∀χ ∈ Σ} for a uniquely determined set Σ of indecomposable elements of ΞX called the spherical roots of X. If a prime divisor of X is B-stable but not G-stable, then it is called a colour ; the set of all colours is denoted by ∆X . The discrete valuation νD associated to a colour D defines a functional ρX (νD ) which is also denoted by ρX (D) for brevity. In general the application D 7→ ρX (D) (where D ranges in ∆X ) is not injective. The stabilizer in G of the union of all colours is a parabolic subgroup containing B denoted by PX . A subgroup H of G is spherical if G/H is a spherical G-variety under the action of left multiplication, and it is wonderful if G/H admits an embedding which is a wonderful variety. If H is wonderful then H necessarily has finite index in NG (H) and the wonderful embedding of G/H is unique. Moreover, if a spherical subgroup H is equal to NG (H) then it is wonderful ([Kn]). 2.2. Relations between spherical roots and colours. If X is a wonderful variety, the set of its spherical roots ΣX is a basis of ΞX . In this setting one can also retrieve ΣX in the following way: let z ∈ X be the unique fixed point of B − and consider the orbit Z = G.z which is the unique closed orbit in X. Then the spherical roots are exactly the T -weights appearing in the quotient Tz X/Tz Z. One can associate to each spherical root γ a G-stable prime divisor Dγ such that γ is the T -weight of Tz X/Tz Dγ . Consider the intersection of all G-invariant prime divisors of X different from Dγ : this intersection is a wonderful variety of rank 1, having γ as its spherical root. The set of spherical roots of all wonderful G-varieties of rank 1 is denoted by Σ(G); our last considerations amount to say that ΣX ⊂ Σ(G) for any wonderful G-variety X. For G of adjoint type the elements of Σ(G) are always linear combinations of simple roots with nonnegative integer coefficients.

WONDERFUL VARIETIES OF TYPE D

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Table 1: Rank one prime wonderful G-varieties for G adjoint of type A D, up to isomorphism. Spherical root

Type

α1

a1

α1 + . . . + αn

an

e q e qpp p ppppp pppppppqp p p p p pp pqpppp ppppppp p ppppqe pe

2α1

a′1

q e

α1 + α′1

a1 × a1 (or d2 ) d3

α1 + 2α2 + α3 2α1 + . . . + 2αn−2 + αn−1 + αn

dn

Diagram

q e

pppppppppppqe

q

q

SL(2) / GL(1) SL(n + 1) / GL(n), n ≥ 2 SL(2) / NG (GL(1))

q e ppppppppppppppqe

q

G/H

q

(SL(2) × SL(2)) / (SL(2) · CG ), SL(2) embedded diagonally SL(4) / (Sp(4) · CG )

q q SO(2n) / (SO(2n − 1) · CG ), @ @q n ≥ 4

For any given G there exist only finitely many wonderful varieties of rank 1, and they are classified ([A], [B2]). With this regard it is convenient to recall the following. ◦ ∼ Definition 2.2. A wonderful variety X with open G-orbit XG = G/H is said to be prime if H satisfies the following conditions:

- if P is a parabolic subgroup such that P r ⊆ H ⊆ P then P = G; - if G = G1 × G2 and H = H1 × H2 with Hi ⊆ Gi then G1 = G or G2 = G. See Section 3.1 for a motivation of this definition. We report in Table 1 the list of rank one prime wonderful G-varieties for G an adjoint group of type A D, up to isomorphism. Two G-varieties, X and X ′ , are here considered to be isomorphic if there exists an outer automorphism σ of G such that X is G-isomorphic to X ′ endowed with the action of G through σ. How to read Table 1. For any rank one prime wonderful G-variety X we report, ◦ in Column 4, the homogeneous space G/H ∼ and, moreover, the following = XG informations: (Column 1) the spherical root as linear combination of simple roots denoted as usually, (Column 2) the type of the spherical root (the type will be used throughout the text to refer to the corresponding spherical root according with this table), (Column 3) the Luna diagram of the spherical system of X. Rank two prime wonderful varieties have been classified by Wasserman ([W]); in Table 2 we report the ones for adjoint groups of type A D (up to isomorphism, as in Table 1). How to read Table 2. For any rank two prime wonderful G-variety X we re◦ port, in Column 3, the homogeneous space G/H ∼ and, moreover: (Column 1) = XG the label according with [W] and the Luna diagram of the spherical system, (Column 2) the set of spherical roots and the set of colours with the table of values taken by the functional associated to each colour on the spherical roots. When necessary, in Column 3, we describe the subgroup H by providing its connected center C ◦ , the commutator subgroup (L, L) of a Levi subgroup L and the Lie

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algebra of its unipotent radical H u in a rather concise way: in a matrix we report the generic element (c1 , . . . , crank Ξ(H) ) of C ◦ times the generic element of (L, L), plus the generic element of Lie(H u ) delimited by a line. We denote with π the usual isogeny π : SL(2) × SL(2) → SO(4) and with j the usual injection j : SO(2n − 1) → SO(2n). Table 2: Rank two prime wonderful G-varieties for G adjoint of type A D, up to isomorphism. G / H (C ◦ , (L, L), Lie(H u )) SL(3) / (SO(3) · CG )

Σ, δαi (γj )

Diagram A1

q e

{2α1 , 2α2 },

q e

′ δα 1 ′ δα 2

A2

2 −1

−1 2

{α1 + α′1 , α2 + α′2 },

q e

q e

q e

q e

δα1 = δα′ 1 δα2 = δα′

2

A3

ppppppppppppppqe

q

ppppppppppppppqe

q

−1 2

{α1 + 2α2 + α3 , α3 + 2α4 + α5 }

q

δα2 δα4

A4(i)

2 −1

δα1 = δα3 + δα 2 − δα 2

qe

A4(ii)

q e

q e

qe

A4(iii)

2 −1 −1

A5

qe

δα1 = δαn δα2 δαn−1

SL(4) / NG (SL(2) × SL(2))

2 −1

−2 2

2 −1 −1

q e

qe

δα1

= δα3 + δα 2 − δα 2

SL(n + 1) / S(GL(2) × GL(n − 1))

−1 1 1

SL(4) / H,

{α1 + α3 , α2 }

e q e

SL(6) / (Sp(6) · CG )

−1 1 1

{α1 + αn , α2 + . . . + αn−1 }

p qpeppp p pp pp p pp pp p pqp pp p p p pppqpppp pp p pp pp p pp p pqe

SL(3) embedded diagonally

SL(4) / S(GL(2) × GL(2))

{α1 + α3 , 2α2 } δα1 = δα3 ′ δα 2

(SL(3) × SL(3)) / (SL(3) · CG ),

−1 2

{α1 + α3 , α2 }

e q e

q e

q e

2 −1

2 0 −2

−1 1 1

cA M

0 c

−1

A

for A ∈ SL(2), c ∈ GL(1), M = M ′ + M ′′ ∈ sl(2) ⊕ C, M ′′ = 0

A6(i)

e q e

A6(ii)

e q e

SL(3) / H,

{α1 , α2 }

e q e

+ δα 1 − δα 1 + δα 2 − δα 2

1 1 0 −1

0 −1 1 1

{α1 , α2 + . . . + αn }

qp p pp pp p pp pp p pp ppqp pp p p p pppqppp p pp p pp pp p pp ppqe e

+ δα 1 − δα 1 δα2

δαn

1 1 −1 0

0 −1 1 1



c1  0 ∗

0 c2 0

for c1 , c2 ∈ GL(1)

 0 0  ∗

SL(n + 1) / H, 

c1  0 ∗

0 ∗ 0

 0 0  c2 A

for A ∈ SL(n − 1), c1 , c2 ∈ GL(1)


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Table 2: Rank two prime wonderful G-varieties (continuation). Σ, δαi (γj )

Diagram A6(iii)

qpepppp pp p pp pp p p p pppp p pp pp p pqe

q pqppp pp p pp pp p p p ppp p pp pp p ppe e

{α1 +. . .+αm , αm+1 +. . .+ αn } δα1 δαm

+ δα

m+1 − δα n

A7(i)

1 1 −1

0 −1 1

0

1

A7(ii)

δα1 δα2 δα3

0 ∗ 0

 0  0 c 2 A2

for A1 ∈ SL(m), A2 ∈ SL(n − m − 1), c1 , c2 ∈ GL(1)

1 1 −2

1 −2 1

1 1 −1



c  m ∗

0 1 −m

0 0 c−1

for c ∈ GL(1), m ∈ C

 

SL(4) / H,

{α1 + α2 , α2 + α3 }

ppp p ppppp pppppppe q qpep pppp p ppppp ppppqe

c 1 A1 0 ∗

SL(3) / H,

+ + = δα δα 2 1 − δα 1 − δα 2

e q e

 

{α1 , α2 }

e q e

G / H (C ◦ , (L, L), Lie(H u )) SL(n + 1) / H,



c  M1 ∗

−1 1 1

0 A M2

0 0 c−1

 

for A ∈ SL(2), c ∈ GL(1), M1 , M2 ∈ C2 ∼ = (C2 )∗ , M1 + M2 = 0

A8

(SL(2) × SL(2)) / B − ,

{α1 , α2 }

e q e

+ δα 1

e q e

D1(i)

q

e q e

ppppppppppppppqe

q

q

q @ @q

D1(ii)

q ppppppppppppppqe

q e

q

q

q @ @q

D3

=

+ δα 2 − δα 1 − δα2

1 1 −1

1 −1 1

{α1 , 2α2 + . . . + 2αn−2 + αn−1 + αn } + δα 1 − δα 1 δα2

1 1 −1

2 −2

SL(2)

embedded diagonally

SO(2n) / (GL(1) × SO(2n − 2))

−1 −1 2

{2α1 , 2α2 + . . . + 2αn−2 + αn−1 + αn } ′ δα 1 δα2

B − Borel subgroup of

SO(2n) / (NSL(2) (GL(1)) × SO(2n − 2))

−1 2

{α1 +2α2 +α3 , α3 +α4 +α5 }

SO(10) / GL(5)

qe q

pppppppqe pppp

δα2 δα4 δα5

q @ @e q

D4(i)

qe qpppppp pp p pp p pp ppqp pp p p p pppqppp p pp pp p pp pp p ppqe e @ @e q

2 −1 −1

−1 1 1

{α1 + . . . + αn−2 , αn−1 + αn }

δαn−1

δα1 δαn−2 = δαn

1 1 −1

0 −2 2

SO(2n) / H,  

c A1 M ∗

0 π(A2 , A2 ) ∗

 0 0  ∗

for A1 ∈ SL(n − 2), A2 ∈ SL(2), c ∈ GL(1), M = M ′ + M ′′ ∈ (Cn−2 )∗ ⊗ C4 ∼ = (Cn−2 )∗ ⊗ n−2 ∗ (sl(2)⊕C) ∼ ((C ) ⊗(sl(2))⊕ = (Cn−2 )∗ ), M ′′ = 0

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Table 2: Rank two prime wonderful G-varieties (continuation). Σ, δαi (γj )

Diagram D4(ii)

{α1 , 2α2 + . . . + 2αn−2 + αn−1 + αn }

q

e q e

ppppppppppppppqe

q

q

+ δα 1 − δα 1 δα2

q @ @q

1 1 −1

0 −2 2

G / H (C ◦ , (L, L), Lie(H u )) SO(2n) / H, 

c  M ∗

 0 0  ∗

0 j(A) ∗

for A ∈ SO(2n − 3), c ∈ GL(1), M = M ′ + M ′′ ∈ C2n−2 ∼ = C2n−3 ⊕ C, M ′′ = 0

D4(iii)

q qpepp pp p pppp p p p p pppp pp p pp ppqe

ppppppppppppppqe

q @ @q

{α1 + . . . + αm , 2αm+1 + . . . + 2αn−2 + αn−1 + αn } δα1 δαm

δαm+1

1 1 −1

0 −2 2

SO(2n) / H,  

c A1 M ∗

0 j(A2 ) ∗

 0 0  ∗

for A1 ∈ SL(m), A2 ∈ SO(2n − 2m − 1), c ∈ GL(1), M = M ′ + M ′′ ∈ (Cm )∗ ⊗C2n−2 ∼ = (Cm )∗ ⊗ m ∗ 2n−3 (C2n−3 ⊕C) ∼ ((C ) ⊗C )⊕ = (Cm )∗ , M ′′ = 0

D7

{α1 + . . . + αn−2 + αn−1 , α1 + . . . + αn−2 + αn }

qe ppppppppppp pp pppp pp pppp pppp pp ppqpp pppp pp pp pp ppppppqpppp pppp pppp pp pppp pppp pp pppp pp q qe @ @qe

δα1 + δα n−1 − δα n

1 1

1 −1

−1

1

SO(2n) / H, cA  M  −M ∗ 

0 1 0 ∗

0 0 1 ∗

 0 0  0  ∗

for A ∈ SL(n − 1), c ∈ GL(1), M ∈ (Cn−1 )∗

Now let S be the set of simple roots associated to B and let X be a wonderful G-variety. For every α ∈ S, let P{α} be the standard parabolic subgroup associated to α; let ∆X (α) denote the set of non-P{α} -stable colours. For brevity we say that α “moves” the colours in ∆X (α), and we should point out that a colour is always moved by some simple root. Lemma 2.3 ([L2]). For all α ∈ S, ∆X (α) has at most two elements. The following four distinct cases can occur. (1) ∆X (α) = ∅, this happens exactly when the open B orbit is stable under P{α} . p p = P The set of all such α is denoted by SX ; we have obviously PSX X. (2) ∆X (α) has two elements, this happens exactly when α ∈ ΣX . The two colours in ∆X (α) are denoted by Dα+ , Dα− , and we have: hρ(Dα+ ), γi + hρ(Dα− ), γi = hα∨ , γi for every γ ∈ ΣX . We denote by AX the union of ∆X (α) for α ∈ S ∩ ΣX . (3) ∆X (α) has one element and 2α ∈ ΣX . The colour in ∆X (α) is denoted by Dα′ , and: 1 hρ(Dα′ ), γi = hα∨ , γi 2 for every γ ∈ ΣX . (4) The remaining case, i.e. ∆X (α) has one element, but 2α ∈ / ΣX . In this case, the colour in ∆X (α) is denoted by Dα , and: hρ(Dα ), γi = hα∨ , γi for every γ ∈ ΣX .


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Lemma 2.4 ([L3]). For all α, β ∈ S, the condition ∆X (α) ∩ ∆X (β) 6= ∅ occurs only in the following two cases: - if α, β ∈ S ∩ ΣX then it can happen that the cardinality of ∆X (α) ∪ ∆X (β) is equal to 3, - if α and β are orthogonal and α + β (or 12 (α + β)) belongs to ΣX , then Dα = Dβ . The relations appearing in these two lemmas have been found using some analysis of the cases in rank 1 and 2, and they will appear in the next section as axioms for an abstract combinatorial structure called spherical system. These objects take p into account the set SX of simple roots moving no colour, the set ΣX of spherical roots and the subset of colours AX . 2.3. Spherical systems. Definition 2.5. Let (S p , Σ, A) be a triple such that S p ⊂ S, Σ ⊂ Σ(G) and A a finite set endowed with an application ρ : A → Ξ∗ , where Ξ = hΣi ⊂ Ξ(T ). For every α ∈ Σ ∩ S, let A(α) denote the set {δ ∈ A : δ(α) = 1}. Such a triple is called a spherical system for G if: (A1) for every δ ∈ A and γ ∈ Σ we have hρ(δ), γi ≤ 1, and if δ(γ) = 1 then γ ∈ S ∩ Σ; (A2) for every α ∈ Σ ∩ S, A(α) contains two elements and by denoting with δα+ and δα− these elements, it holds hρ(δα+ ), γi + hρ(δα− ), γi = hα∨ , γi, for every γ ∈ Σ; (A3) the set A is the union of A(α) for all α ∈ Σ ∩ S; (Σ1) if 2α ∈ Σ ∩ 2S then 12 hα∨ , γi is a nonpositive integer for every γ ∈ Σ \ {2α}; (Σ2) if α, β ∈ S are orthogonal and α + β ∈ Σ (or 12 (α + β) ∈ Σ) then hα∨ , γi = hβ ∨ , γi for every γ ∈ Σ; (S) for every γ ∈ Σ, there exists a wonderful G-variety X of rank 1 with γ as p spherical root and S p = SX . The cardinality (rank) of Σ is the rank of the spherical system. Using the finite list of rank one prime wonderful varieties, in case the group G is of type A D the last axiom can be rewritten as follows: (S) (type A D) for every γ ∈ Σ, {α ∈ S : hα∨ , γi = 0} ∩ supp(γ) ⊂ S p ⊂ {α ∈ S : hα∨ , γi = 0}, where the set supp(γ) is the support of γ, namely {α ∈ S : hα∗ , γi = 6 0} (here {α∗ } denotes the dual basis of S). Remark 2.6. For every spherical system the spherical roots are linearly independent and Σ is a basis of Ξ. p On a wonderful variety X the information contained in SX , ΣX and AX is enough to “recover” the full set of colours ∆X at least as an abstract set endowed with an application ρX : ∆X → Ξ∗ . This is easily done using lemmas 2.3 and 2.4; let us see this procedure applied to an abstract spherical system (S p , Σ, A). Its set of colours ∆ is defined as follows: ′

∆ = ∆a ⊔ ∆a ⊔ ∆b where

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∆a = A, ′ ∆a = S ∩ 21 Σ, ∆b = S b /∼ , where S b = S \ (Σ ∪ 21 Σ ∪ S p ) and α ∼ β if and only if α ⊥ β and α + β ∈ Σ (or 21 (α + β) ∈ Σ). The set of elements of ∆ “corresponding” to α ∈ S is denoted by ∆(α), and we say that these are the colours moved by α. The element corresponding to α ∈ S ∩ 12 Σ ′ in ∆a is denoted by δα′ , and the element corresponding to α ∈ S b in ∆b is denoted by δα . The map ρ : A → Ξ∗ can be extended to ∆ as follows: hρ(δα′ ), γi = 12 hα∨ , γi ∀α ∈ S ∩ 12 Σ, ∀γ ∈ Ξ; hρ(δα ), γi = hα∨ , γi ∀α ∈ S b , ∀γ ∈ Ξ. The definition of spherical system is such that the following lemmas hold (see [L3]): p Lemma 2.7. For every wonderful G-variety X the triple (SX , ΣX , AX ) is a spherical system. p Lemma 2.8. The map X 7→ (SX , ΣX , AX ) is a bijection between rank one (resp. rank two) wonderful varieties (up to G-isomorphism) and rank one (resp. rank two) spherical systems.

In [L3] it is proven that spherical systems classify wonderful G-varieties for G adjoint of type A. Our main theorem is the generalization of this result to type A D. Theorem 2.9. Let G be a semisimple adjoint algebraic group of mixed type A and p D. Then the map associating to a wonderful G-variety X the triple (SX , ΣX , AX ) is a bijection between wonderful G-varieties (up to G-isomorphism) and spherical systems for G. Remark 2.10. Obviously, two G-isomorphic wonderful G-varieties have the same spherical system. If two wonderful G-varieties are isomorphic, namely G-isomorphic up to an outer automorphism of G, their spherical systems are equal up to a permutation of the set S of simple roots. A useful way to represent a spherical system for G is provided by the Luna diagram constructed on the Dynkin diagram of the root system of G. We focus on adjoint groups of type A D, following [L3] in type A. For every α ∈ S, we draw one or two circles near the corresponding vertex to represent the colours moved by α. Different circles corresponding to the same colour are joined by a line. In the Luna diagram each spherical root in Σ is denoted as in Table 11 on the vertices corresponding to supp(γ). In order to represent the set A and the map ρ : A → Ξ∗ , we arrange the diagram in such a way that the circle above a simple root α (corresponding to a colour denoted by δα+ ) satisfies hρ(δα+ ), γi ≥ −1 for all γ ∈ Σ. Moreover we draw one arrow, > or 0 ∀δ ∈ ∆′ , δ∈∆′

such that hφ, γi ≥ 0 for all γ ∈ Σ. Let Σ(∆′ ) denote the maximal subset of spherical roots such that there exists a linear combination φ as above such that hφ, γi > 0 for all γ ∈ Σ(∆′ ). The subset ∆′ ⊂ ∆ is distinguished if and only if there exists a vector subspace V ′ of V such that the convex cone generated by ρ(∆′ ) and V ′ ∩ V is equal to V ′ , and V ′ ∩ V is a face of V. If such a subspace V ′ exists, it is unique and it is denoted by V (∆′ ). We have V (∆′ ) = hρ(∆′ ) ∪ {γ ∗ : γ ∈ Σ(∆′ )}i. where {γ ∗ } is the dual basis of Σ.

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Let (S p , Σ, A) be a spherical system and ∆′ a distinguished subset of ∆. Let Ξ/∆′ be the subgroup of Ξ defined by: Ξ ∆′

=

{ξ : hv, ξi = 0 ∀v ∈ V (∆′ )}

=

{ξ : hρ(δ), ξi = 0 ∀δ ∈ ∆′ and hγ ∗ , ξi = 0 ∀γ ∈ Σ(∆′ )}.

The triple (S p /∆′ , Σ/∆′ , A/∆′ ) defined as follows is called the quotient triple of (S p , Σ, A) by ∆′ and it is also denoted by (S p , Σ, A)/∆′ . The set S p /∆′ is {α ∈ S p : ∆(α) ⊂ ∆′ } ⊂ S p . The set Σ/∆′ is the set of the indecomposable elements of the semigroup X { cγ γ ∈ Ξ/∆′ : cγ ≥ 0 ∀γ ∈ Σ}. γ∈Σ

′

The set A/∆ ⊂ A is the union of those A(α) which satisfy A(α) ∩ ∆′ = ∅, ∗ and the map ρ/∆′ : A/∆′ → (Ξ/∆′) is the restriction of ρ followed by the ∗ ′ ∗ projection Ξ → (Ξ/∆ ) . In general, the set Σ/∆′ is not always a subset of Σ. If it is a basis of Ξ/∆′ then the distinguished subset ∆′ is said to have the ∗-property of Luna. In this case the quotient triple is a spherical system. Remark 3.1. It is not clear whether there exist examples of distinguished subsets without ∗-property. All distinguished subsets considered in the following have the ∗-property. Let Φ : X → X ′ be a dominant G-morphism between wonderful G-varieties. Set ∆Φ = {D ∈ ∆X : Φ(D) = X ′ }. Proposition 3.2 ([L3]). Let X be a wonderful G-variety: the application Φ 7→ ∆Φ is a bijection between the set of dominant G-morphisms with connected fibers of X onto another wonderful G-variety, and the set of distinguished subsets of ∆X having the ∗-property. Moreover, for any such Φ : X → X ′ the spherical system of X ′ is the quotient p , ΣX , AX )/∆Φ . triple (SX The distinguished subset ∆′ of ∆ is smooth if the dimension of the space V (∆′ ) is equal to the dimension of the face V (∆′ ) ∩ V of V. Equivalently, if V (∆′ ) = hγ ∗ : γ ∈ Σ(∆′ )i and, hence, Σ/∆′ = Σ \ Σ(∆′ ) ⊂ Σ. In particular, if ∆′ is smooth then it has the ∗-property. Proposition 3.3 ([L3]). Let Φ : X → X ′ be a dominant G-morphism with connected fibers between wonderful G-varieties. Then Φ is smooth if and only if ∆Φ is smooth. The distinguished subset ∆′ of ∆ is parabolic if V (∆′ ) = V , or equivalently if Σ(∆′ ) = Σ, namely Σ/∆′ = ∅. In particular, in this case ∆′ is smooth. Proposition 3.4 ([L3]). Let X be a wonderful G-variety and let S ′ be a subset of S. The map Φ 7→ ∆Φ is a bijection between the set of the G-morphisms Φ : X → G/PS ′ and the set of the parabolic distinguished subsets of ∆X ; the bijection is such that S ′ = S p /∆′ .


11

Parabolic induction. Let P be a parabolic subgroup of G and let L ⊂ P be a Levi subgroup. Let Y be a wonderful L-variety. The parabolic induction obtained from Y by P is the wonderful G-variety G ×P Y , where Y is considered as a P -variety where the radical P r of P acts trivially and G ×P Y is the algebraic quotient of G × Y by the following action of P : p.(g, y) = (g p−1 , p.y), for all p ∈ P , g ∈ G, y ∈ Y. On the other hand, a wonderful variety X is obtained by parabolic induction by P if and only if P r ⊂ H ⊂ P , where G/H is the open G-orbit of X. Proposition 3.5 ([L3]). Let X be a wonderful G-variety and let S ′ be a subset of S. Let PS−′ denote the parabolic subset containing B − associated to S ′ . Then X is obtained by parabolic induction by PS−′ if and only if (supp(Σ) ∪ S p ) ⊂ S ′ . In this case the G-morphism Φ : X → G/PS−′ is unique and ∆Φ is equal to the parabolic distinguished subset ∆(S ′ ). A spherical system is cuspidal if supp(Σ) = S. In particular, a cuspidal spherical system cannot be obtained by parabolic induction by any proper parabolic subgroup. Direct and fiber product. A spherical system (S p , Σ, A) is said to be reducible if there exists a partition of S into two subsets S1 , S2 such that: S1 ⊥ S2 , ∀γ ∈ Σ supp(γ) ⊂ S1 or supp(γ) ⊂ S2 , ∀δ ∈ A(Σ ∩ S1 ), ρ(δ) is null on S2 , and vice versa. In this case (S p , Σ, A) is the direct product of the spherical systems (S p ∩ S1 , Σ1 , A(Σ ∩ S1 )) and (S p ∩ S2 , Σ2 , A(Σ ∩ S2 )), where Σi = {γ ∈ Σ : supp(γ) ⊂ Si }. For this notion of reducibility, the obvious statement of “unique factorization” is true. Let (S p , Σ, A) be a spherical system. Let ∆1 and ∆2 be two distinguished subsets of ∆, obviously ∆3 = ∆1 ∪ ∆2 is distinguished. The subsets ∆1 and ∆2 decompose into fiber product the spherical system (S p , Σ, A) if: (i) (ii) (iii) (iv) (v)

∆1 6= ∅, ∆2 6= ∅ and ∆1 ∩ ∆2 = ∅, ∆1 , ∆2 and ∆3 have the ∗-property, (Σ \ (Σ/∆1 )) ∩ (Σ \ (Σ/∆2 )) = ∅, ((S p /∆1 ) \ S p ) ⊥ ((S p /∆2 ) \ S p ), ∆1 or ∆2 is smooth.

Notice that Σ \ (Σ/∆′ ) = Σ(∆′ ) when ∆′ is smooth. Proposition 3.6 ([L3]). Let (S p , Σ, A) be a spherical system, let ∆1 and ∆2 be two distinguished subsets that decompose (S p , Σ, A). Let us suppose that, for i = 1, 2, 3, there exists a wonderful G-variety Xi unique up to G-isomorphism with spherical system (S p , Σ, A)/∆i . Then there exists two G-morphisms Φ1 : X1 → X3 , Φ : X2 → X3 such that the fiber product X1 ×X3 X2 is a wonderful G-variety with spherical system (S p , Σ, A) and every wonderful G-variety with this spherical system is Gisomorphic to X1 ×X3 X2 . Notice that the above operation of factorization is a particular case of this decomposition into fiber product. A reducible spherical system, S = S1 ⊔ S2 , is decomposed by ∆(S1 ) and ∆(S2 ), and it corresponds to a direct product of two wonderful varieties.

12


Projective fibration. Let X and Y be two wonderful G-varieties. A G-morphism Φ : X → Y is called a projective fibration if it is smooth, all its fibers are isomorphic to a projective space of fixed dimension n and rank(X) = rank(Y ) + n. We can look at the spherical systems in order to characterize these morphisms Φ in terms of conditions on ∆Φ . Let (S p , Σ, A) be a spherical system. An element δ of A is projective if hρ(δ), γi ≥ 0 for all γ ∈ Σ. Notice that this condition is equivalent to hρ(δ), γi ∈ {0, 1}, thanks to axiom A1. Let δ ∈ A be a projective element, let Sδ denote the set {α ∈ Σ ∩ S : hρ(δ), αi = 1}. The subset {δ} is distinguished and smooth, the quotient triple (S p , Σ, A)/{δ} is a spherical system where: S p /{δ} = S p , Σ/{δ} = Σ \ Sδ , A/{δ} = A(S \ Sδ ), and ρ/{δ} is the restriction of ρ followed by the projection on (Ξ/{δ})∗ . Proposition 3.7. Let G be of type A D. Let (S p , Σ, A) be a spherical system with a projective element δ ∈ A. Let us suppose that there exists a wonderful G-variety Y unique up to G-isomorphism with spherical system (S p , Σ, A)/{δ}. Then there exists a wonderful G-variety X unique up to G-isomorphism with spherical system (S p , Σ, A) and if Φδ : X → Xδ denotes the G-morphism such that ∆Φδ = {δ} we have that Xδ is G-isomorphic to Y . In [L3] the above proposition is stated only for the type A, but its proof holds also in the case of type A D. Indeed, it is enough to notice that the key lemma (3.6.2, loc. cit.) is true whenever no spherical root α being also a simple root lies in the support of any other spherical root γ ∈ Σ. This occurs in type A as well as it occurs in type A D, as can be seen from the list of rank two spherical systems. 3.2. Proof of Theorem 2.9: Reduction. In order to prove that for all spherical systems (S p , Σ, A) for G adjoint of type A D there exists a wonderful G-variety unique up to a G-isomorphism, it is sufficient to prove that this holds for all spherical systems called primitive, that are: - cuspidal, - without projective elements, - indecomposable, and in particular irreducible. Indeed, we can proceed by induction on the rank of S and on the rank of Σ and we can use propositions 3.5, 3.6 and 3.7 about parabolic inductions, fiber products and projective fibrations. The list of primitive spherical systems is contained in Table 3. Such a list can be obtained by a combinatorial reduction, that is explained in the next two paragraphs: we report some technical definitions and outline the strategy, similar to that of [L3]. How to read Table 3. For any primitive wonderful G-variety X we report the label and the Luna diagram of the spherical system and the homogeneous space G/H ∼ = ◦ . When necessary we describe the subgroup H, as in Table 2, by providing its XG connected center C ◦ , the commutator subgroup (L, L) of a Levi subgroup L and the Lie algebra of its unipotent radical H u . In some cases, where the component d(ni ) occurs, we use further special notations: if ni = 2 (with d(2) we denote aa(1, 1)) φi (A) = π(A, a) for A ∈ SL(2), π : SL(2) × SL(2) → SO(4) isogeny and Vi ∼ = sl(2);


13

if ni > 2 φi (A) = j(A) for A ∈ SO(2ni − 1), j : SO(2ni − 1) → SO(2ni ) injection and Vi ∼ = C2ni −1 . Table 3: Primitive wonderful G-varieties with rank at least three for G adjoint of type A D, up to isomorphism. Diagram, H, (C ◦ , (L, L), Lie(H u )) 1. ao(n).

G

q e

q e

q e

q e

pppppppppppqe

q

q

pppppppqe pppp

qe

qpp ppppppp p p p p p pp ppppp ppppe e q

SL(n + 1)

SO(n + 1) · CG 2. ac(n), n ≥ 3 odd.

SL(n + 1)

q Sp(n + 1) · CG 3. aa(p + q + p).

qe

q SL(2p + q + 1)

e q

e q

S(GL(p + q) × GL(p + 1)) 4. aa(p, p). Localization of aa(p + q + p) in S \ {αp+1 , . . . , αp+q }. SL(p + 1) · CG , with SL(p + 1) embedded diagonally 5. aa∗ (p + 1 + p).

q e

qe

q e

e q

SL(p + 1) × SL(p + 1)

SL(2p + 2)

e q

NG (SL(p + 1) × SL(p + 1)) 6. ac∗ (n), n ≥ 3.

SL(n + 1)

p pp pp p pp qpepppp pp pp p pp pp p ppqep pp pp p pp pp p pp ppqe

ppp p pp pp pqepp pp p pp pp p pp pp pe qpp p pp pp p pp pp p pp pe q

n even: NG (Sp(n)). nodd:  c  M1 ∗

0 A M2

0 0  ∗

for A ∈ Sp(n−1), c ∈ GL(1), M1 , M2 ∈ Cn−1 ∼ = (Cn−1 )∗ , M1 + M2 = 0. 7. ax(1 + p + 1 + q + 1). SL(p + q + 4)

e q e 

c 1 A1 0 ∗

0 c 2 A2 0

qpeppppp p pp pp p p p p pppp pp pp p ppqe

e q e

qpp p pp pp p pp p p p p pppppppp pp p pqe e

e q e

 0  0 c 3 A3

for A1 ∈ SL(p + 1), A2 ∈ SL(2), A3 ∈ SL(q + 1), c1 , c2 , c3 ∈ GL(1), c1 c2 c3 = 1. 8. ax(1 + p + 1, 1). SL(p + 3) × SL(2) Localization of ax(1 + p + 1 + q + 1) in S \ {αp+3 , . . . , αp+q+2 }. NG (SL(p + 1) × SL(2)), with SL(2) embedded diagonally. 9. ax(1 + p + 1). SL(p + 3) Localization of ax(1 + p + 1 + q + 1) in {α , . . . , α }. 1 p+2   



c 1 A1 0 0

0 c2 ∗

0 0  ∗

for A1 ∈ SL(p + 1), c1 , c2 ∈ GL(1).

10. ax(1, 1, 1). SL(2) × SL(2) × SL(2) Localization of ax(1 + p + 1 + q + 1) in {α1 , αp+2 , αp+q+3 }. SL(2) · CG , with SL(2) embedded diagonally.

14


Table 3: Primitive wonderful G-varieties (continuation). Diagram, H, (C ◦ , (L, L), Lie(H u ))

G

11. ay(p + q + p), p ≥ 2.

e q e  

c 1 A1 0 0

0 c 2 A2 ∗

SL(2p + q + 1)

e q e

e q e  0 0  ∗

e q e

e q e

qpp p p ppppp p p p p pp ppppppp pp e e qp

e q e

e q e

e q e

for A1 ∈ SL(p+q), A2 ∈ SL(p+1), c1 , c2 ∈ GL(1).

12. ay(p + q + (p − 1)), p ≥ 2. Localization of ay(p + q + p) in S \ {α2p+q }.   ∗  ∗ 0

0 c 1 A1 0

0  0 c 2 A2

SL(2p + q)

for A1 ∈ SL(p+q−1), A2 ∈ SL(p), c1 , c2 ∈ GL(1).

13. ay(p, p), p ≥ 2. SL(p + 1) × SL(p + 1) Localization of ay(p + q + p) in S \ {α , . . . , α }. p+1 p+q cA 0 cA 0 , for A ∈ SL(p), c ∈ GL(1), M ∈ (Cp )∗ . M ∗ −M ∗ 14. ay(p, (p − 1)), p ≥ 2. Localization of ay(p, p) in S \ {α′p }. NG (SL(p)), with SL(p) embedded diagonally. 15. ay(n), n ≥ 4 even.



cA  M1 ∗

e q e

e q e

e q e

e q e

e q e

SL(p + 1) × SL(p)

SL(n + 1)

e q e

e q e

e q e

 0  0 c−1 tA−1

0 1 M2

for A ∈ SL(n/2), c ∈ GL(1), M1 , M2 ∈ (Cn/2 )∗ , M1 + M2 = 0. 16. ay∼ (p + q + p), p ≥ 2. SL(2p + q + 1)

e q e    

c 1 A1 M1 ∗ ∗

0 ∗ 0 M2

e q e

0 0 c 2 A2 ∗

 0  0   0 t −1 c 3 A1

17. ay∗ (2 + q + 2).

e q e c1  ∗  M 1 ∗ 

0 c 2 A1 0 ∗

e q e

e q e

0 0 c 3 A2 M2

 0 0  0  ∗

qpp pp ppppppp p p p pp p p pppppp e e qp

e q e

e q e

e q e

e q e

for A1 ∈ SL(p), A2 ∈ SL(q), c1 , c2 , c3 ∈ GL(1), 2p+1 c2p+1 c2q = 1, M1 , M2 ∈ (Cp )∗ , M1 +M2 = 0. 1 2 c3

e q e

SL(q + 5)

e qp pp pp p pp p pp p p p pppppp p pp pp e qp

e q e

e q e

for A1 ∈ SL(q + 1), A2 ∈ SL(2), c1 , c2 , c3 ∈ GL(1), cq2 c33 = 1, M1 , M2 ∈ C2 ∼ = (C2 )∗ , M1 + M2 = 0.


15


G

18. az∼ (3 + q + 3).

SL(q + 7)

e q e c1  M1   ∗  ∗   ∗ ∗

0 c 2 A1 M2 ∗ ∗ ∗

∗  M1  ∗   ∗ ∗

0 c 1 A1 M2 ∗ ∗



0 0 2 c−1 1 c2 0 M3 ∗

e q e

e q e

0 0 0 c 3 A2 ∗ ∗

0 0 0 0 3 c−2 1 c 2 A1 M4

0 0 0 0 0 ∗

      

e q e

e q e

qp pp pp p pp p pp p p p pppppp p pp pp e e qp

for A1 ∈ SL(2), A2 ∈ SL(q), c1 , c2 , c3 ∈ GL(1), c82 cq3 = c31 , M1 , M2 , M3 , M4 ∈ C2 ∼ = (C2 )∗ , M1 + M2 + M3 + M4 = 0.

19. az∼ (3 + q + 2). Localization of az ∼ (3 + q + 3) in S \ {αq+6 }. 

0 0 c2 0 M3

0 0 0 c 3 A2 ∗

0 0 0 0 −1 2 c 1 c 2 A1

    

e q e

SL(q + 6)

for A1 ∈ SL(2), A2 ∈ SL(q), c1 , c2 , c3 ∈ GL(1), c63 cq3 = c21 , M1 , M2 , M3 ∈ C2 ∼ = (C2 )∗ , M1 + M2 + M3 = 0.

20. az(3, 3). SL(4) × SL(4) Localization of az ∼ (3 + q + 3) in {α1 , α2 , α3 , αq+4 , αq+5 , αq+6 }.     for A ∈ SL(2), c ∈ GL(1), c 0 0 c 0 0   M1 A 0  ,  M3 A 0   M1 , M2 , M3 , M4 ∈ C2 ∼ = (C2 )∗ , ∗ M2 c−1 ∗ M4 c−1 M1 + M2 + M3 + M4 = 0. 21. az(3, 2). SL(4) × SL(3) ′ Localization of az(3, 3) in S \ {α }. 3   −2   −2 c 0 0 c 0   M1   for A ∈ SL(2), c ∈ GL(1), M1 , M2 , M3 ∈ A 0 , M3 cA 2 C2 ∼ = (C2 )∗ , M1 + M2 + M3 = 0. ∗ M2 c 22. az(3, 1). SL(4) × SL(2) Localization of az(3, 3) in S\ {α′1 , α′3 }.   c 0 0 for A ∈ SL(2), c ∈ GL(1), M1 , M2 ∈ C2 ∼ = (C2 )∗ ,   M1 A 0  , (A) −1 M + M = 0. 1 2 ∗ M2 c 23. ae6 (6).

SL(7)

e q e

e q e c1  M1   ∗  ∗ ∗

0 c2 A M2 ∗ ∗

c−5  M1   ∗ ∗

0 c−2 A M2 ∗



0 0 2 c−1 1 c2 M3 ∗

0 0 0 −2 3 c1 c2 A M4

0 0 0 0 ∗

24. ae6 (5). Localization of ae6 (6) in S \{α6 }.  0 0 c M3

0 0   0  4 c A

    

e q e

e q e

e q e

e q e

for A ∈ SL(2), c1 , c2 ∈ GL(1), c71 = c10 2 , M1 , M2 , M3 , M4 ∈ C2 ∼ = (C2 )∗ , M1 + M2 + M3 + M4 = 0. SL(6)

for A ∈ SL(2), c ∈ GL(1), M1 , M2 , M3 ∈ C2 ∼ = (C2 )∗ , M1 + M2 + M3 = 0.

16



G

25. ae7 (7).

SL(8)

c3  M1   ∗ ∗

0 cA M2 ∗

∗  M1 ∗

0 c1 A M2



0 0 −1 t −1 c A M3



0 0 0 c−3

e q e

e q e

e q e   

e q e

e q e

e q e

e q e

for A ∈ SL(3), c ∈ GL(1), M1 , M3 ∈ C3 , V M2 = M2′ + M2′′ ∈ (Sym2 C3 )∗ ⊕ ( 2 C3 )∗ , V 2 C3 ∼ = ( C3 )∗ , M1 + M2′′ + M3 = 0.

26. ae7 (6). Localization of ae7 (7) in S \ {α7 }. 

SL(7)

0  0 A−1

for A ∈ SL(3), c1 , c2 ∈ GL(1), c51 c22 = 1, M1 ∈ C3 , M2 = V M2′ + M2′′ ∈ (Sym2 C3 )∗ ⊕ ( 2 C3 )∗ , M1 + M2′′ = 0. c2 27. ae7 (5). SL(6) Localization of ae7 (7) in S \ {α1 , α7 }. for A ∈ SL(3), c ∈ GL(1), M = M ′ + M ′′ ∈ (Sym2 C3 )∗ ⊕ cA 0 V −1 t −1 M c A ( 2 C3 )∗ , M ′′ = 0. 28. af (5). SL(6) t

e q e

e q e 

c A1  M1 ∗

0 A2 M2

0 0 c

−1

A1

 

e q e

e q e

e q e

for A1 , A2 ∈ SL(2), c ∈ GL(1), M1 , M2 ∈ C2 ⊗ (C2 )∗ ∼ = (C2 )∗ ⊗ C2 , M1 + M2 = 0.

29. af (4). Localization of af (5) in S \  {α5 }.  c1  ∗   M1 ∗

0 c−4 2 0 ∗

0 0 c2 A M2

0 0 0

2 c−1 1 c2

30. do(p + q), q ≥ 3.

  

SL(5)

for A ∈ SL(2), c1 , c2 ∈ GL(1), M1 , M2 ∈ C2 ∼ = (C2 )∗ , M1 + M2 = 0.

q q e

q e

q e

q e

ppppppppppppppqe

q

SO(2p + 2q)

q @ @q

S(O(p + 1) × O(2q + p − 1)) 31. do(p + 2).

SO(2p + 4)

qe q e

q e

q e

S(O(p + 1) × O(p + 3)) 32. do(n).

q e NG (SO(n) × SO(n))

q e

q e

q e @ @e q

q e

q e @ @qq e

SO(2n)


17


G

33. dc(n), n even.

ppppppppppppppqe

q

q

ppppppppppppppqe

q

SO(2n)

q ppppppppppe q pppp e @ @q e

q

GL(n) 34. dc′ (n), n even.

SO(2n)

q pppppppppppqe

q

q

pppppppppppqe

q

q pppppppe pppp @ @q e

q

NG (GL(n)) 35. dc(n), n odd.

SO(2n)

qe pppppppppppqe

q

q

q

pppppppqe pppp

q

pppppppqe pppp

q @ @e q

GL(n) 36. dd(p, p), p ≥ 4.

q e

SO(2p) × SO(2p)

qe

qe

qe qe @ @qe

q e

qe

qe

qe qe @ @qe

SO(2p) · CG , with SO(2p) embedded diagonally. 37. ds∗ (4).

SO(8)

e q e qppppppppppp pp pppp pppp pp pppp pp q @ @e q G2 · CSO(8) 38. dc∗ (n)

SO(2n)

qe p pp ppppp p ppppp ppppppp p ppqe qpepppppppp pppppp pqe n odd:

cA  M  −M ∗

neven: 2 c  M1   ∗   ∗  ∗ 0

0 1 0 ∗ 0 cA M2 M3 ∗ ∗

0 0 ∗ ∗ 0 0 1 0 ∗ ∗

 0 0  0  ∗ 0 0 0 ∗ ∗ ∗

ppppppp ppe qp pppp p pppppp p pp pe q @ @e q

for A ∈ Sp(n − 1), c ∈ GL(1), M ∈ (Cn−1 )∗ . 0 0 0 0 ∗ ∗

0 0 0 0 0 ∗

      

for A ∈ Sp(n − 2), c ∈ GL(1), M1 , M2 , M3 ∈ Cn−2 ∼ = (Cn−2 )∗ , M1 + M2 + M3 = 0.

18



G

39. dy(p, p), p ≥ 4.

cA   M1  M 2 ∗

0 1 0 ∗

0 0 ∗ ∗

e q e

e q e @ e @q e

e q e

e q e

e q e



SO(2p) × SO(2p)

  0 cA 0   M3 ,  0 M4 ∗ ∗

0 1 0 ∗

cp A M2 M3 ∗

0 0 ∗ ∗

0 0 ∗ ∗

40. dy(p, (p − 1)), p ≥ 4. Localization of dy(p, p) in S \ {αp }.   

cA M1

0 c1−p



 ,

0 1 0 ∗

 0 0  0  ∗

 0 0  0  ∗

e q e

e q e

e q e

 cA  M1   (A) ,  M 2 ∗

0 1 0 ∗

0 0 ∗ ∗

 0 0  0  ∗

42. dy(7).

for A ∈ SL(p − 1), c ∈ GL(1), M1 , M2 , M3 ∈ (Cp−1 )∗ , M1 +M2 +M3 = 0.

     

c5 A M1 ∗ ∗ ∗ ∗

0 c3 M2 ∗ 0 ∗

0 0 c tA−1 M3 ∗ ∗

0 0 0 ∗ ∗ ∗

0 0 0 0 ∗ ∗

(SL(p − 1) × SO(2p))

for A ∈ SL(p − 1), c ∈ GL(1), M1 , M2 ∈ (Cp−1 )∗ , M1 + M2 = 0. SO(14)

e q e

e q e



0 0 0 0 0 ∗

43. dz(4, 4).

      

e q e

e q e @ e @q e

e q e

e q e

for A ∈ SL(3), c ∈ GL(1), M1 , M2 , M3 ∈ (C3 )∗ ∼ = V2 3 C , M1 + M2 + M3 = 0.

SO(8) × SO(8)

e q e

c2   M1   ∗   ∗  ∗ 0

0 cA M2 M3 ∗ ∗

0 0 1 0 ∗ ∗

0 0 0 ∗ ∗ ∗

0 0 0 0 ∗ ∗

0 0 0 0 0 ∗

c2  M4   ∗ ,  ∗  ∗ 0

       

e q e

e q e @ e @q e

e q e



e q e @ e @q e

for A ∈ SL(p − 1), c ∈ GL(1), M1 , M2 , M3 , M4 ∈ (Cp−1 )∗ , M1 + M2 + M3 + M4 = 0. SL(p) × SO(2p)

41. dy(p, (p − 2)), p ≥ 4. Localization of dy(p, p) in S \ {αp−1 , αp }. 

e q e

e q e @ e @q e

e q e

0 cA M5 M6 ∗ ∗

0 0 1 0 ∗ ∗

0 0 0 ∗ ∗ ∗

0 0 0 0 ∗ ∗

0 0 0 0 0 ∗

      

for A ∈ SL(2), c ∈ GL(1), M1 , . . . , M6 ∈ C2 ∼ = (C2 )∗ , M1 +. . .+ M6 = 0.


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44. dz(4, 3). Localization of dz(4, 4) in S \ {α4 }. 

 c2   M3  ∗  ,   ∗  ∗ 0

 c   M  1  ∗ 

0 A M2



c−6   M2  0 , ∗ cA  ∗  ∗ 0

0 0 c−1

SL(4) × SO(8)

0 cA M4 M5 ∗ ∗

0 0 1 0 ∗ ∗

0 0 0 ∗ ∗ ∗

0 0 0 0 ∗ ∗

0 0 0 0 0 ∗

45. dz(4, 2). Localization of dz(4, 4) in S \ {α3 , α4 }. 0 c−3 A M3 M4 ∗ ∗



   c−2  M1  

0 0 1 0 ∗ ∗

0 0 0 ∗ ∗ ∗

0 0 0 0 ∗ ∗

0 0 0 0 0 ∗

46. dz(4, 1). Localization of dz(4, 4) in S \ {α1 , α3 , α4 }.  c2  M1     ∗   (A) ,   ∗   ∗  0 

0 cA M2 M3 ∗ ∗

0 0 1 0 ∗ ∗

0 0 0 ∗ ∗ ∗

0 0 0 0 ∗ ∗

0 0 0 0 0 ∗

47. de6 (7).

      

c4  M1   ∗   ∗   ∗   ∗   ∗   ∗  ∗ 0 

0 c3 A M2 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 c2 M3 ∗ ∗ ∗ 0 ∗ ∗

0 0 0 cA M4 M5 ∗ ∗ ∗ ∗

0 0 0 0 1 0 ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 0 ∗ ∗ ∗ ∗

         

c3 A M1 ∗ ∗ ∗ ∗ ∗ ∗

0 c2 M2 ∗ ∗ ∗ 0 ∗

0 0 cA M3 M4 ∗ ∗ ∗

0 0 0 1 0 ∗ ∗ ∗

0 0 0 0 ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ ∗ ∗

0 0 0 0 0 0 ∗ ∗

e q e 0 0 0 0 0 0 0 ∗ ∗ ∗

0 0 0 0 0 0 0 ∗

for A ∈ SL(2), c ∈ GL(1), M1 , . . . , M5 ∈ C2 ∼ = (C2 )∗ , M1 + . . . + M5 = 0. SL(3) × SO(8)

for A ∈ SL(2), c ∈ GL(1), M1 , . . . , M4 ∈ C2 ∼ = (C2 )∗ , M1 + . . . + M4 = 0. SL(2) × SO(8)

SO(14)

48. de6 (6). Localization of de6 (7) in S \ {α1 }. 

     

     

for A ∈ SL(2), c ∈ GL(1), M1 , M2 , M3 ∈ C2 ∼ = (C2 )∗ , M1 + M2 + M3 = 0.

e q e

e q e





0 0 0 0 0 0 0 0 ∗ ∗

e q e

e q e @ e @q e

e q e 0 0 0 0 0 0 0 0 0 ∗

              

for A ∈ SL(2), c ∈ GL(1), M1 , . . . , M5 ∈ C2 ∼ = (C2 )∗ , M1 + · · · + M5 = 0.

SO(12)

          

for A ∈ SL(2), c ∈ GL(1), M1 , . . . , M4 ∈ C2 ∼ = (C2 )∗ , M1 + · · · + M4 = 0.

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49. de7 (8).

SO(16)

c3  M1   ∗   ∗   ∗   ∗  ∗ 0 

0 c2 A M2 ∗ ∗ ∗ ∗ ∗

0 0 c tA−1 M3 M4 ∗ ∗ ∗

0 0 0 1 0 ∗ ∗ ∗

0 0 0 0 ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ ∗ ∗

0 0 0 0 0 0 ∗ ∗

0 0 0 0 0 0 0 ∗

50. de7 (7). Localization of de7 (8) in S \ {α1 }.       

c2 A M1 ∗ ∗ ∗ ∗

0 c tA−1 M2 M3 ∗ ∗

0 0 1 0 ∗ ∗

0 0 0 ∗ ∗ ∗

0 0 0 0 ∗ ∗

0 0 0 0 0 ∗

51. de8 (8).

      

          

e q e @ e @q e

e q e

for A ∈ SL(3), c ∈ GL(1), M1 , M3V , M4 ∈ C3 , M2 = M2′ + ′′ M2 ∈ ( 2 C3 )∗ ⊕ (Sym2 C3 )∗ , C3 ∼ = V ( 2 C3 )∗ , M1 + M2′ + M3 + M4 = 0.

SO(14)

for A ∈ SL(3), c ∈ GL(1), M2 , M3 ∈ C3 , V M1 = M1′ + M1′′ ∈ ( 2 C3 )∗ ⊕ (Sym2 C3 )∗ , V 2 C3 ∼ = ( C3 )∗ , M1′ + M2 + M3 = 0.

e q e

e q e

e q e

e q e

e q e

e q e

e q e

e q e

e q e

SO(16)

e q e

e q e @ e @q e

e q e

for A ∈ SL(4), c ∈ GL(1), M1 ∈ C4 , M2 = V2 M2′ +M2′′ ∈ C6 ⊗(C4 )∗ ∼ = ( C4 )⊗(C4 )∗ ∼ =V⊕ C4 , M1 + M2′′ = 0 (π : SL(4) → SO(6) isogeny, V V Cartan product of ( 2 C4 ) and (C4 )∗ ). 52. de8 (7). SO(14) Localization of de8 (8) in S \ {α1 }.   for A ∈ SL(4), c ∈ GL(1), M = M ′ + M ′′ ∈ C6 ⊗ (C4 )∗ ∼ = cA 0 0 V2 4 t  M C ) ⊗ (C4 )∗ ∼ = V ⊕ C4 , MV′′ = 0 (π : SL(4) → SO(6) π(A)−1 0  ( ∗ ∗ ∗ isogeny, V Cartan product of ( 2 C4 ) and (C4 )∗ ). c2  M1   ∗  ∗ 0 

0 cA M2 ∗ ∗

0 0 t π(A)−1 ∗ ∗

0 0 0 ∗ ∗

0 0 0 0 ∗

    

53. aa(1) + ds(n).

SO(2n + 2)

qe

e q e   

c1 0 0 0

0 c2 A ∗ 0

0 0 ∗ 0

 0 0  0  ∗

qpppppppppppppppppppppppp e

ppp p pp pp p p q pppppppp p @ @qe

for A ∈ SL(n), c1 , c2 ∈ GL(1).

54. aa(1) + ds∗ (4).

SO(10)

e q e SO(2) × Spin(7)

e q pqpppppppppp pppppppppppppppppp q e @ @e q


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55. ac∗ (n) + aa(1), n ≥ 3.

SO(2n + 2)

e q ppppppp ppe qpp p pp pp p pp pp p pp pe q e @ @qq e

pp p pp p pp pppp pp p pp pp p pp p pqe p pp p pp pp p pp pp p ppqe qe   

c1 0 ∗ 0

0 c2 A 0 ∗

0 0 ∗ 0

 0 0  0  ∗

for A ∈ SL(n), c1 , c2 ∈ GL(1).

56. 2−comb + ac∗ (3).

SO(10)

e q e

e q e c32  ∗   M1   ∗  ∗ 0 

0 c1 0 ∗ 0 ∗

0 0 c2 A M2 ∗ ∗

0 0 0 ∗ 0 ∗

0 0 0 0 ∗ ∗

0 0 0 0 0 ∗

      

qpepp p ppppp pppppppqe @ @e q

for A ∈ SL(3), c1 , c2 ∈ GL(1), M1 , M2 ∈ C3 ∼ = V ( 2 C3 )∗ , M1 + M2 = 0.

57. ac∗ (n1 ) + d(n2 ), n1 ≥ 3, n2 ≥ 2.

pp pp p pp p e qpppppppp p pp pp p ppe qppppp pp p pp p pp pp pqe n1 even:

n1 odd:

 

c A1 M ∗

c2  M1   ∗  ∗ 0 

 0 0  ∗

0 φ(A2 ) ∗ 0 c A1 M2 ∗ ∗

0 0 φ(A2 ) ∗ ∗

pppppppp pe qpppp p pp pp p pp pp p pe qp pp p pp p pp pp p pp pe qp

e q e

    

c 1 A1 0 M ∗ 0

0 c 2 A2 ∗ ∗ ∗

0 0 φ(A3 ) ∗ ∗

0 0 0 ∗ ∗

0 0 0 0 ∗

    

0 0 0 0 ∗

    

 

c A1 M1 ∗

0 φ1 (A2 ) ∗

e q e

  0 c A1 0  ,  M2 ∗ ∗

ppppppppppppppqe

q @ @q

for A1 ∈ SL(p + q), A2 ∈ SL(p), c1 , c2 ∈ GL(1), M = M ′ +M ′′ ∈ C2n ⊗(Cp+q )∗ ∼ = (V ⊗(Cp+q )∗ )⊕ (Cp+q )∗ , M ′′ = 0.

q ppppppppppppppqe

q

e q e

e q e

59. ay(p, p) + d(n1 ) + d(n2 ), p, n1 , n2 ≥ 2.

e q e

q @ @q

for A1 ∈ Sp(n1 − 1), c ∈ GL(1), M1 ∈ Cn1 −1 , M2 = M2′ + M2′′ ∈ C2n2 ⊗ (Cn1 −1 )∗ ∼ = (V ⊗ (Cn1 −1 )∗ ) ⊕ Cn1 −1 , M1 + ′′ M2 = 0. SO(4p + 2q + 4)

pppppp pp pp p p p p ppp pp pp p pp pqe qe 0 0 0 ∗ 0

ppppppppppqe pppp

SO(2n1 + 2n2 )

for A1 ∈ Sp(n1 ), c ∈ GL(1), M = M ′ + M ′′ ∈ C2n2 ⊗ (Cn1 )∗ ∼ = (V ⊗ (Cn1 )∗ ) ⊕ (Cn1 )∗ , M ′′ = 0.

58. ay(p + q + p) + d(n), p, n ≥ 2.

e q e

q

q @ @q 0 φ2 (A3 ) ∗

SO(2p + 2n1 ) × SO(2p + 2n2 )

e q e  0 0  ∗

e q e

q ppppppppppppppe q

q @ @q

for A1 ∈ SL(p), c ∈ GL(1), M1 = M1′ + M1′′ ∈ C2n1 ⊗ (Cp )∗ ∼ = (V1 ⊗ (Cp )∗ ) ⊕ (Cp )∗ , ′ ′′ 2n2 p ∗ ∼ p ∗ p ∗ ′′ M2 = M2 + M2 ∈ C ⊗ (C ) = (V2 ⊗ (C ) ) ⊕ (C ) , M1 + M2′′ = 0.

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60. ay(p, p) + d(n), p, n ≥ 2. SL(p + 1) × SO(2p + 2n) Localization of ay(p + q + p) + d(n) in S \ {αp+1 , . . . , αp+n1 }.    2 for A1 ∈ SL(p), c ∈ GL(1), M1 ∈ (Cp )∗ , c A1 0 0 c A1 0     , M2 φ(A2 ) 0 M2 = M2′ + M2′′ ∈ C2n ⊗ (Cp )∗ ∼ = (V ⊗ M1 c−2 ∗ ∗ ∗ (Cp )∗ ) ⊕ (Cp )∗ , M1 + M2′′ = 0. 61. ay(p, (p − 1)) + d(n), p, n ≥ 2. SL(p) × SO(2p + 2n) Localization of ay(p, p) + d(n) in S \ {α }. p    c A1 M ∗

0 φ(A2 ) ∗

0 0  ∗

for A1 ∈ SL(p), c ∈ GL(1), M = M ′ + M ′′ ∈ C2n ⊗ (Cp )∗ ∼ = (V ⊗ (Cp )∗ ) ⊕ (Cp )∗ , M ′′ = 0. 62. ay(p + q + (p − 1)) + d(n), p, n ≥ 2. SO(4p + 2q + 2n − 2)  (A1 ) , 

e q e

e q e     

c 1 A1 0 M ∗ ∗

0 c 2 A2 ∗ ∗ ∗

0 0 φ(A3 ) ∗ ∗

qpepppp p pp pp p p p p ppp pp pp p pp pqe 0 0 0 ∗ 0

0 0 0 0 ∗



63. dy(p + q + (p − 1)), p ≥ 3.

      

c 1 A1 0 M1 M2 ∗ ∗

0 c 2 A2 ∗ ∗ ∗ ∗

0 0 1 0 ∗ ∗

e q e

e q e

e q e 0 0 0 1 ∗ ∗

0 0 0 0 ∗ 0

   

0 0 0 0 0 ∗

      

e q e

e q e

for A1 ∈ SL(p), A2 ∈ SL(p+q−1), c1 , c2 ∈ GL(1), M = M ′ + M ′′ ∈ C2n ⊗ (Cp )∗ ∼ = (V ⊗ (Cp )∗ ) ⊕ (Cp )∗ , M ′′ = 0. SO(4p + 2q − 2)

e qpp p pp pp p pp p p p p pppppp pp p pp e qp

e q e

e q e @ e @q e

e q e

for A1 ∈ SL(p+q), A2 ∈ SL(p), c1 , c2 ∈ GL(1), M1 , M2 ∈ (Cp+q )∗ , M1 + M2 = 0.

64. dy(p, (p − 1)) + d(n), p ≥ 4, n ≥ 2.

e q e

  

c A1 M1 M2 ∗

0 1 0 ∗

0 0 1 ∗

e q e

e q e @ e @qq e   0 c A1 0   M3 , 0  ∗ ∗

q q @ @q

pppppppppppqe

e q e

0 φ(A2 ) ∗

SO(2p)×SO(2p+2n−2)

e q e

 0  0  ∗

e q e

q ppppppppppppppqe

q @ @q

for A1 ∈ SL(p), c ∈ GL(1), M1 , M2 ∈ (Cp )∗ , M3 = M3′ +M3′′ ∈ C2n ⊗(Cp )∗ ∼ = (V ⊗(Cp )∗ )⊕(Cp )∗ , M1 +M2 +M3′′ = 0.


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65. dy′ (p + q + (p − 2)), p ≥ 4.

e q e

e q e       

c 1 A1 0 M1 M2 ∗ ∗

0 c 2 A2 ∗ ∗ ∗ ∗

0 0 1 0 ∗ ∗

0 0 0 1 ∗ ∗

0 0 0 0 ∗ 0

SO(4p + 2q − 4)

0 0 0 0 0 ∗

      

e q e @ e @q e

e q e

e q e

qpepp p pp pp p pp p p p pp p pp p pp pp pqe

e q e

for A1 ∈ SL(p), A2 ∈ SL(p + q − 1), c1 , c2 ∈ GL(1), M1 , M2 ∈ (Cp )∗ , M1 + M2 = 0.

66. ay∗ (2 + q + 2) + d(n), n ≥ 2.

e q e c22  ∗   M1   ∗   ∗  ∗ 0 

0 c 1 A1 0 ∗ ∗ ∗ ∗

0 0 c 2 A2 M2 ∗ ∗ ∗

e q e

SO(4p + 2q + 4)

qpepppp p p pppp p p p ppp p pppppp pqe

0 0 0 φ(A3 ) ∗ ∗ ∗

0 0 0 0 ∗ 0 ∗

0 0 0 0 0 ∗ ∗

67. dy∗ (3 + q + 3).

e q e

e q e

c32  ∗   M1   ∗  ∗ 0 

0 c 1 A1 0 ∗ ∗ ∗

0 0 c 2 A2 M2 ∗ ∗

0 0 0 ∗ 0 ∗

e q e

0 0 0 0 ∗ ∗

0 0 0 0 0 ∗

e q e

      

0 0 0 0 0 0 ∗

        

 

c 1 A1 0 ∗ ∗

0 c 2 A2 0 ∗

0 0 ∗ 0

 0 0  0  ∗

q pppppppqe pppp

q @ @q

for A1 ∈ SL(q + 1), A2 ∈ SL(2), c1 , c2 ∈ GL(1), M1 ∈ C2 , M2 = M2′ +M2′′ ∈ C2n ⊗ (C2 )∗ ∼ = (V ⊗(C2 )∗ )⊕(C2 ), M1 +M2′′ = 0. SO(2q + 12)

e q e

qpepppp ppppppp p p p ppppp ppppppqe

e q e @ e @q e

for A1 ∈ SL(q +2), A2 ∈ SL(3), c1 , c2 ∈ GL(1), V2 M1 , M2 ∈ C3 ∼ = ( C3 )∗ , M1 + M2 = 0.

68. dy∗ (3 + q + 2). Localization of dy ∗ (3 + q + 3) in S \ {α1 }. 

e q e

SO(2q + 10)

for A1 ∈ SL(q +2), A2 ∈ SL(3), c1 , c2 ∈ GL(1).

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69. dy∼ (4 + q + 3).

SO(2q + 10)

          

c52 A1 ∗ M1 ∗ ∗ ∗ ∗ ∗

0 c 1 A2 0 ∗ ∗ ∗ ∗ ∗

0 0 c32 M2 ∗ 0 ∗ ∗

e q e

e q e

e q e

0 0 0 c2 tA−1 1 M3 ∗ ∗ ∗

0 0 0 0 ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ 0 ∗

0 0 0 0 0 0 ∗ ∗

0 0 0 0 0 0 0 ∗

          

70. ay(2, 2) + 2−comb.

e q e

c31  M1   ∗   ∗  ∗ 0 

0 c 1 A1 0 ∗ M2 ∗

0 0 c 2 A2 ∗ ∗ ∗

0 0 0 ∗ 0 ∗

0 0 0 0 ∗ ∗

0 0 0 0 0 ∗

      

e q e

pqppppppp p pp p p p p pppppp pp p pp e e qp

e q e @ e @q e

for A1 ∈ SL(3), A2 ∈ SL(q), c1 , c2 ∈ GL(1), M1 , M2 , M3 ∈ V2 3 (C3 )∗ ∼ C , M1 +M2 +M3 = 0. = SO(12)

e q e

e q e @ e @q e

e q e

e q e

e q e

for A1 ∈ SL(3), A2 ∈ SL(2), c1 , c2 ∈ GL(1), V2 M1 , M2 ∈ C3 ∼ = ( C3 )∗ , M1 + M2 = 0.

71. ay(2, 1) + 2−comb. Localization of ay(2, 2) + 2−comb in S \ {α1 }.   

c 1 A1 0 ∗ 0

0 c 2 A2 ∗ ∗

0 0 ∗ 0

 0 0  0  ∗

SO(10)

for A1 ∈ SL(3), A2 ∈ SL(2), c1 , c2 ∈ GL(1).

72. az(3, 3) + d(n1 ) + d(n2 ), n1 , n2 ≥ 2.

e q e c2   M1   ∗  ∗ 0 

0 c A1 M2 ∗ ∗

e q e

q

e q e

0 0 φ1 (A2 ) ∗ ∗

ppppppppppppppqe 0 0 0 ∗ ∗

0 0 0 0 ∗

q @ @q c2  M3  , ∗  ∗ 0

     

SO(2n1 + 6) × SO(2n2 + 6)

0 c A1 M4 ∗ ∗

e q e

e q e

e q e

0 0 φ2 (A3 ) ∗ ∗

q q @ @q

q ppppppppppppppe 0 0 0 ∗ ∗

0 0 0 0 ∗

    

for A1 ∈ SL(2), c ∈ GL(1), M1 , M3 ∈ C2 , M2 = M2′ + M2′′ ∈ C2n1 ⊗ (C2 )∗ ∼ = (V1 ∼ = 2 ∗ 2 ′ ′′ 2n2 2 ∗ ∼ 2 ∗ 2 ′′ (C ) )⊕C , M4 = M4 +M4 ∈ C ⊗(C ) = (V2 ⊗(C ) )⊕C , M1 +M2 +M3 +M4′′ = 0. 73. az(3, 2) + d(n1 ) + d(n2 ), n1 , n2 ≥ 2. SO(2n1 + 6) × SO(2n2 + 4) Localization of az(3, 3) + d(n1 ) + d(n2 ) in S \ {α′1 }. c2   M1   ∗  ∗ 0 

0 c A1 M2 ∗ ∗

0 0 φ1 (A2 ) ∗ ∗

0 0 0 ∗ ∗

0 0 0 0 ∗



 c A1    M3 ,  ∗

0 φ2 (A3 ) ∗

  0   0   ∗

for A1 ∈ SL(2), c ∈ GL(1), M1 ∈ C2 , M2 = M2′ +M2′′ ∈ C2n1 ⊗(C2 )∗ ∼ = (V1 ⊗(C2 )∗ )⊕ 2 ∗ 2 ′′ C2 , M3 = M3′ + M3′′ ∈ C2n2 ⊗ (C2 )∗ ∼ (V ⊗ (C ) ) ⊕ C , M + M + M3′′ = 0. = 2 1 2


25


G

74. az(3, 3) + d(n), n ≥ 2. SL(4) × SO(2n + 6) Localization of az(3, 3) + d(n1 ) + d(n2 ) in S \ {α4 , . . . , αn1 +3 }.     for A1 ∈ SL(2), c ∈ GL(1), c2 0 0 0 0   c 0 0  M3 c A1  0 0 0   M1 , M2 , M3 ∈ C2 ∼ = (C2 )∗ ,  M  A1 0  , ′ ′′ M4 φ(A2 ) 0 0    ∗  1 M = M4 + M4 ∈ C2n ⊗  ∗  0 M2 c−1 ∗ ∗ ∗ 0   42 ∗ (C ) ∼ = (V ⊗ (C2 )∗ ) ⊕ (C2 )∗ , 0 ∗ ∗ ∗ ∗ M1 + M2 + M3 + M4′′ = 0. 75. az(3, 2) +1 d(n), n ≥ 2. SL(3) × SO(2n + 6) Localization of az(3, 3) + d(n) in S \ {α1 }.    c6 0 0 0 0 for A1 ∈ SL(2), c ∈ GL(1),  M2 c3 A1  c A   M , M ∈ C2 ∼ (C2 )∗ , M = 0 0 0 0 = 1 2 3    1 , ∗  M3 φ(A2 ) 0 0   ′ ′′ 2n M1 c−2  ∗  ⊗ (C2 )∗ ∼ = (V ⊗ ∗ ∗ ∗ 0   M3 + M3 ∈ C 0 ∗ ∗ ∗ ∗ (C2 )∗ )⊕(C2 )∗ , M1 +M2 +M3′′ = 0. 2 76. az(3, 2) + d(n), n ≥ 2. SL(4) × SO(2n + 4) Localization of az(3, 3) + d(n) in S \ {α′1 }.      for A1 ∈ SL(2), c ∈ GL(1), M1 , M2 ∈ c 0 0 c A1 0 0 C2 ∼ = (C2 )∗ , M3 = M3′ + M3′′ ∈ C2n ⊗   M1 A1 0  ,  M3 φ(A2 ) 0   2 ∗ ∼ (C ) = (V ⊗ (C2 )∗ ) ⊕ C2 , M1 + M2 + ∗ ∗ ∗ 0 M2 c−1 M3′′ = 0. 77. az(3, 1) + d(n), n ≥ 2. SL(2) × SO(2n + 6) Localization of az(3, 3) + d(n) in S \ {α1 , α3 }.  c2  M1     (A1 ) ,  ∗  ∗  0

0 c A1 M2 ∗ ∗



0 0 φ(A2 ) ∗ ∗

0 0 0 ∗ ∗

0 0 0 0 ∗

78. az∼ (3 + q + 3) + d(n), n ≥ 2.

e q e

e q e                 

c42 M1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0

0 A1 M2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

c32

0 0 c22 0 M3 ∗ ∗ ∗ 0 ∗ ∗

e q e 0 0 0 c 1 A2 ∗ ∗ ∗ ∗ ∗ ∗ ∗

    

for A1 ∈ SL(2), c ∈ GL(1), M1 ∈ C2 , M2 = M2′ + M2′′ ∈ C2n ⊗ (C2 )∗ ∼ = (V ⊗ (C2 )∗ ) ⊕ C2 , ′′ M1 + M2 = 0. SO(2q + 2n + 12)

qpeppppp pp pp p p p p ppp pp pp p pp pqe 0 0 0 0 c 2 A1 M4 ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 φ(A3 ) ∗ ∗ ∗ ∗ ∗

e q e

e q e

e q e 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 0 0 ∗ 0 ∗ ∗

0 0 0 0 0 0 0 0 ∗ ∗ ∗

79. az∼ (3 + q + 2) + d(n), n ≥ 2. Localization of az ∼ (3 + q + 3) + d(n) in S \ {α1 }.             

c32 A1 M1 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 c22 0 M2 ∗ ∗ ∗ 0 ∗

0 0 c 1 A2 ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 c 2 A1 M3 ∗ ∗ ∗ ∗

0 0 0 0 φ(A3 ) ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ ∗ ∗ ∗

0 0 0 0 0 0 ∗ 0 ∗

0 0 0 0 0 0 0 ∗ ∗

0 0 0 0 0 0 0 0 ∗

            

q q @ @q

ppppppppppppppqe 0 0 0 0 0 0 0 0 0 ∗ ∗

0 0 0 0 0 0 0 0 0 0 ∗

                

for A1 ∈ SL(2), A2 ∈ SL(q), c1 , c2 ∈ GL(1), M1 , M2 , M3 ∈ C2 ∼ = (C2 )∗ , M4 = M4′ + M4′′ ∈ C2n ⊗ (C2 )∗ ∼ = (V ⊗ (C2 )∗ ) ⊕ (C2 )∗ , M1 +M2 +M3 +M4′′ = 0. SO(2q + 2n + 10)

for A1 ∈ SL(2), A2 ∈ SL(q), c1 , c2 ∈ GL(1), M1 , M2 ∈ C2 ∼ = (C2 )∗ , M3 = M3′ + M3′′ ∈ C2n ⊗ (C2 )∗ ∼ = (V ⊗(C2 )∗ )⊕C2 , M1 + M2 + M3′′ = 0.

26



G

80. dz(4, 3) + d(n), n ≥ 2.

SO(8) × SO(2n + 6)

e q e e q e @ e @qq e

e q e

c2   M1   ∗   ∗  ∗ 0 

0 c A1 M2 M3 ∗ ∗

0 0 1 0 ∗ ∗

0 0 0 1 ∗ ∗

0 0 0 0 ∗ ∗

0 0 0 0 0 ∗

      



  , 

e q e

e q e

c2 M4 ∗ ∗ 0

e q e

0 c A1 M5 ∗ ∗

q q @ @q

pppp pppppppqe

0 0 φ(A2 ) ∗ ∗

0 0 0 ∗ ∗

0 0 0 0 ∗





     

for A1 ∈ SL(2), c ∈ GL(1), M1 , M2 , M3 , M4 ∈ C2 ∼ = (C2 )∗ , M5 = M5′ + M5′′ ∈ 2n 2 ∗ ∼ 2 ∗ 2 ∗ C ⊗ (C ) = (V ⊗ (C ) ) ⊕ (C ) , M1 + M2 + M3 + M4 + M5′′ = 0. 81. dz(4, 2) + d(n), n ≥ 2. SO(8) × SO(2n + 4) Localization of dz(4, 3) + d(n) in S \ {α′1 }.    for A ∈ SL(2), c ∈ GL(1), 1 c2 0 0 0 0 0  2 ∼ 2 ∗   M1 c A1 0 0 0 0   c A1 0 0    M1 , M2 , M3 ∈ C = (C ) ,  ∗  M2 1 0 0 0   M4 φ(A2 ) 0   M4 = M4′ + M4′′ ∈ C2n ⊗  , M3 0 1 0 0   ∗  ∗ ∗ ∗  ∗  (C2 )∗ ∼ ∗ ∗ ∗ ∗ 0  = (V ⊗ (C2 )∗ ) ⊕ (C2 )∗ , 0 ∗ ∗ ∗ ∗ ∗ M1 + M2 + M3 + M5′′ = 0. ∼ 82. dz (4 + q + 3). SO(2q + 14)

e q e                   

c42 M1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0

0 A1 ∗ M2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

c32

0 0 c 1 A2 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

e q e

e q e

0 0 0 c22 M3 ∗ ∗ ∗ 0 ∗ ∗ ∗

0 0 0 0 0 1 0 ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 0 1 ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗

83. dz∼ (4 + q + 2). Localization of dz ∼ (4 + q + 3) in S \ {α1 }.               

c32 A1 ∗ M1 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 c 1 A2 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 c22 M2 ∗ ∗ ∗ 0 ∗ ∗

0 0 0 c 2 A1 M3 M4 ∗ ∗ ∗ ∗

0 0 0 0 1 0 ∗ ∗ ∗ ∗

0 0 0 0 0 1 ∗ ∗ ∗ ∗

0 0 0 0 0 0 ∗ ∗ ∗ ∗

0 0 0 0 0 0 0 ∗ 0 ∗

0 0 0 0 0 0 0 0 ∗ 0 ∗ ∗

0 0 0 0 0 0 0 0 ∗ ∗

e q e @ e @q e

e q e

pqppppppp pppp p p p ppppp ppppppe e qp

0 0 0 0 c 2 A1 M4 M5 ∗ ∗ ∗ ∗ ∗

e q e

0 0 0 0 0 0 0 0 0 ∗ ∗ ∗

0 0 0 0 0 0 0 0 0 ∗

              

0 0 0 0 0 0 0 0 0 0 ∗ ∗

0 0 0 0 0 0 0 0 0 0 0 ∗

                  

for A1 ∈ SL(2), A2 ∈ SL(q), c1 , c2 ∈ GL(1), M1 , M2 , M3 , M4 , M5 ∈ C2 ∼ = (C2 )∗ , M1 + M2 + M3 + M4 + M5 = 0.

SO(2q + 12)

for A1 ∈ SL(2), A2 ∈ SL(q), c1 , c2 ∈ GL(1), M1 , M2 , M3 , M4 ∈ C2 ∼ = (C2 )∗ , M1 + M2 + M3 + M4 = 0.


27


G

84. dz(3 + q + 2).

SO(2q + 10)

e q e

e q e c22  ∗   M1   ∗   ∗  ∗   ∗ 0 

0 c 1 A1 0 ∗ ∗ ∗ ∗ ∗

0 0 c 2 A2 M2 M3 ∗ ∗ ∗

0 0 0 1 0 ∗ ∗ ∗

e q e

0 0 0 0 1 ∗ ∗ ∗

0 0 0 0 0 ∗ 0 ∗

e q e @ e @q e

qpeppppppp p pp p p p p pppppp pp p pp qe p

0 0 0 0 0 0 ∗ ∗



0 0 0 0 0 0 0 ∗

         

for A1 ∈ SL(q + 1), A2 ∈ SL(2), c1 , c2 ∈ GL(1), M1 , M2 , M3 ∈ C2 ∼ = (C2 )∗ , M1 + M2 + M3 = 0.

85. dz(3 + q + 1). Localization of dz(3 + q + 2) in S \ {α1 }.       

c 1 A1 0 ∗ ∗ ∗ ∗

0 c 2 A2 M1 M2 ∗ ∗

0 0 1 0 ∗ ∗

0 0 0 1 ∗ ∗

0 0 0 0 ∗ 0

0 0 0 0 0 ∗

86. az(3, 2) + 2−comb.

      

c31  M1   ∗   ∗   ∗   ∗   ∗   ∗  ∗ 0 

0 c21 c2

A M2 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 c1 c22 M3 ∗ ∗ ∗ 0 ∗ ∗

0 0 0 c1 M4 ∗ 0 ∗ ∗ ∗

for A1 ∈ SL(q + 1), A2 ∈ SL(2), c1 , c2 ∈ GL(1), M1 , M2 ∈ (C2 )∗ , M1 + M2 = 0. SO(14)

0 0 0 0 c2 A M5 ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 0 ∗ ∗ ∗ ∗

e q e

e q e @ e @q e

e q e

e q e

e q e

e q e

SO(2q + 8)

0 0 0 0 0 0 0 ∗ ∗ ∗

0 0 0 0 0 0 0 0 ∗ ∗

0 0 0 0 0 0 0 0 0 ∗

87. ae6 (6) + d(n), n ≥ 2.

e q e c4  M1   ∗   ∗   ∗   ∗   ∗  ∗ 0 

0 c 3 A1 M2 ∗ ∗ ∗ ∗ ∗ ∗

0 0 c2 M3 ∗ ∗ 0 ∗ ∗

0 0 0 c A1 M4 ∗ ∗ ∗ ∗

e q e

e q e 0 0 0 0 φ(A2 ) ∗ ∗ ∗ ∗

0 0 0 0 0 ∗ ∗ ∗ ∗

e q e

e q e

e q e 0 0 0 0 0 0 ∗ ∗ ∗

0 0 0 0 0 0 0 ∗ ∗

0 0 0 0 0 0 0 0 ∗

              

for A ∈ SL(2), c1 , c2 ∈ GL(1), M1 , M2 , M4 ∈ C2 ∼ = (C2 )∗ , V 2 2 M3 , M5 ∈ C ∼ = ( C )∗ , M1 + M2 + M4 = 0, M3 + M5 = 0. SO(2n + 12)

q ppppppppppqe pppp             

q @ @q

for A1 ∈ SL(2), c ∈ GL(1), M1 , M2 , M3 ∈ C2 ∼ = (C2 )∗ , M4 = M4′ + M4′′ ∈ C2n ⊗ (C2 )∗ ∼ = (V ⊗ (C2 )∗ ) ⊕ (C2 )∗ , M1 + M2 + M3 + M4′′ = 0.

28



G

88. ae6 (5) + d(n), n ≥ 2. Localization of ae6 (6) in S \ {α1 }.         

c 3 A1 M1 ∗ ∗ ∗ ∗ ∗

0 c2 M2 ∗ ∗ 0 ∗

0 0 c A1 M3 ∗ ∗ ∗

0 0 0 φ(A2 ) ∗ ∗ ∗

0 0 0 0 ∗ ∗ ∗

SO(2n + 10)

0 0 0 0 0 ∗ ∗

0 0 0 0 0 0 ∗

89. ae7 (7) + d(n), n ≥ 2.

c3  M1   ∗   ∗   ∗  ∗ 0 

0 c 2 A1 M2 ∗ ∗ ∗ ∗

0 0 t −1 c A1 M3 ∗ ∗ ∗

0 0 0 φ(A2 ) ∗ ∗ ∗

0 0 0 0 ∗ ∗ ∗

       

for A1 ∈ SL(2), c ∈ GL(1), M1 , M2 ∈ C2 ∼ = (C2 )∗ , M3 = M3′ + M3′′ ∈ C2n ⊗ 2 ∗ ∼ (C ) = (V ⊗ (C2 )∗ ) ⊕ (C2 )∗ , M1 + M2 + M3′′ = 0. SO(2n + 14)

e q e

e q e

e q e

e q e

e q e

e q e



0 0 0 0 0 ∗ ∗

0 0 0 0 0 0 ∗

        

e q e

   

c 2 A1 M1 ∗ ∗ ∗

0 c tA−1 1 M2 ∗ ∗

0 0 φ(A2 ) ∗ ∗

0 0 0 ∗ ∗

0 0 0 0 ∗

91. 3−comb + aa(q).

     

c1 A 0 M1 M2 ∗ ∗

0 c2 ∗ ∗ 0 ∗

0 0 1 0 ∗ ∗

0 0 0 1 ∗ ∗

0 0 0 0 ∗ 0

0 0 0 0 0 ∗

92. 3−comb + 2−comb.

      

q @ @q

SO(2n + 12)



for A1 ∈ SL(3), c ∈ GL(1), M1 = M1′ + M1′′ ∈ V V (Sym2 C3 )∗ ⊕( 2 C3 )∗ , ( 2 C3 )∗ ∼ = C3 , M2 = M2′ + ′′ 2n 3 ∼ M2 ∈ C ⊗ C = (V ⊗ C3 ) ⊕ C3 , M1′′ + M2′′ = 0.

   

SO(2q + 6)

e q e qpep pppp p pppp p p p pp pp pppppppe q e @ @qq e

e q e 

pppp pppppppqe

for A1 ∈ SL(3), c ∈ GL(1), M1 ∈ C3 , M2 = M2′ + M2′′ ∈ (Sym2 C3 )∗ ⊕ V V ( 2 C3 ) ∗ , ( 2 C3 ) ∗ ∼ = C3 , M3 = M3′ + ′′ 2n M3 ∈ C ⊗ C3 ∼ = (V ⊗ C3 ) ⊕ C3 , M1 + M2′′ + M3′′ = 0.

90. ae7 (6) + d(n), n ≥ 2. Localization of ae7 (7) in S \ {α1 }. 

q

for A ∈ SL(q+1), c1 , c2 ∈ GL(1), M1 , M2 ∈ (Cq+1 )∗ , M1 + M2 = 0. SO(10)

e q e e q e

e q e c21  M1   ∗   ∗   ∗  ∗   ∗ 0 

0 c1 A 0 M2 M3 ∗ ∗ ∗

0 0 c2 ∗ ∗ 0 ∗ ∗

0 0 0 1 0 ∗ ∗ ∗

0 0 0 0 1 ∗ ∗ ∗

0 0 0 0 0 ∗ 0 ∗

0 0 0 0 0 0 ∗ ∗

0 0 0 0 0 0 0 ∗

e q e @ e @qq e

          

for A ∈ SL(2), c1 , c2 ∈ GL(1), M1 , M2 , M3 ∈ C2 ∼ = (C2 )∗ , M1 + M2 + M3 = 0.


29


G

93. ax(1 + p + 1) + d(n), n ≥ 2.

e q e     

c 1 A1 0 M ∗ ∗

0 c2 ∗ 0 ∗

0 0 φ(A2 ) ∗ ∗

0 0 0 ∗ 0

0 0 0 0 ∗

    

0 c 1 A1 0 M2 ∗ ∗ ∗

0 0 c2 ∗ 0 ∗ ∗

0 0 0 φ(A2 ) ∗ ∗ ∗

0 0 0 0 ∗ 0 ∗

q q @ @q

pppppppppppqe

for A1 ∈ SL(p + 1), c1 , c2 ∈ GL(1), M = M ′ + M ′′ ∈ C2n ⊗ (Cp+1 )∗ ∼ = (V ⊗ (Cp+1 )∗ ) ⊕ (Cp+1 )∗ , M ′′ = 0. SO(2n + 8)

e q e

e q e c21  M1   ∗   ∗   ∗  ∗ 0

e q e

qpep ppppp pppp p p p pp ppppp ppppqe

94. af (4) + d(n), n ≥ 2.



SO(2p + 2n + 4)

0 0 0 0 0 ∗ ∗

e q e

e q e 0 0 0 0 0 0 ∗

        

q ppppppppppppppqe

q @ @q

for A1 ∈ SL(2), c1 , c2 ∈ GL(1), M1 ∈ C2 , M2 = M2′ + M2′′ ∈ C2n ⊗ (C2 )∗ ∼ = (V ⊗ (C2 )∗ ) ⊕ C2 , M1 + M2′′ = 0.

Remark 3.8. Some primitive spherical systems reported in Table 3 are associated to the wonderful embedding of symmetric varieties (see 4.1). In Table 4 we report the corresponding Satake diagrams. Table 4: Symmetric cases. Spherical system, G, H ao(n), G = SL(n + 1), H = SO(n + 1) · CG ac(n), G = SL(n + 1), H = Sp(n + 1) · CG aa(p + q + p), q ≥ 2, G = SL(2p + q + 1), H = S(GL(p + q) × GL(p + 1)) aa(p, p), G = SL(p + 1) × SL(p + 1), H = SL(p + 1) · CG aa(p + 1 + p), G = SL(2p + 2), H = NG (SL(p + 1) × SL(p + 1))

Satake diagram

30


do(p + q), G = SO(2p + 2q), H = S(O(p + 1) × O(2q + p − 1))

do(p + 2), G = SO(2n), H = S(O(n − 1) × O(n + 1))

do(n), G = SO(2n), H = NG (SO(n) × SO(n))

dc(n), n odd, G = SO(2n), H = GL(n)

dc′ (n), n even, G = SO(2n), H = NG (GL(n)

dd(p, p), G = SO(2p) × SO(2p), H = SO(2p) · CG

Strong ∆-connectedness. Let (S p , Σ, A) be a spherical system and let ∆ be its set of colours. Two spherical roots γ1 , γ2 ∈ Σ are said to be strongly ∆-adjacent ([L3]) if for all δ ∈ ∆({γ1 }) we have hρ(δ), γ2 i = 6 0 and, vice versa, for all δ ∈ ∆({γ2 }) we have hρ(δ), γ1 i = 6 0. See Column 2 of Table 2 and notice that, in type A D, two spherical roots with overlapping supports are strongly ∆-adjacent. Let Σ′ be a subset of Σ. The localization (S p ∩ supp(Σ′ ), Σ′ , A(S ∩ Σ′ )) of p (S , Σ, A) in Σ′ is said to be strongly ∆-connected if for every couple of spherical roots γ1 , γ2 ∈ Σ′ there exists a finite sequence of spherical roots in Σ′ , one strongly ∆-adjacent to the next one, starting with γ1 and ending with γ2 . If Σ′ ⊂ Σ is


31

maximal with this property we say that (S p ∩ supp(Σ′ ), Σ′ , A(S ∩ Σ′ )) is a strongly ∆-connected component 2 of (S p , Σ, A). Let (S p , Σ, A) be a spherical system. Let Σ′ be a subset of spherical roots. Let ∆(Σ′ ) denote the subset of colours δ ∈ ∆(supp(Σ′ )) such that hρ(δ), γi = 0 for all γ ∈ Σ \ Σ′ . We say that the localization (S p ∩ supp(Σ′ ), Σ′ , A(S ∩ Σ′ )) of (S p , Σ, A) is erasable 3 if there exists a nonempty smooth distinguished subset of colours ∆′ included in ∆(Σ′ ). Notice that such a subset is smooth and distinguished with respect to the spherical system (S p , Σ, A) if and only if it is smooth and distinguished with respect to the localization (S p ∩ supp(Σ′ ), Σ′ , A(S ∩ Σ′ )). Weakening this requirement, we say that the localization (S p ∩ supp(Σ′ ), Σ′ , A(S ∩ Σ′ )) is quasi-erasable if there exists a nonempty distinguished subset of colours ∆′ included in ∆(Σ′ ) and satisfying the ∗-property. Lemma 3.9. Let (S p , Σ, A) be a spherical system. Let Σ1 and Σ2 be two disjoint subsets of Σ giving two quasi-erasable localizations such that at least one of them is erasable. Then the corresponding subsets ∆′1 ⊂ ∆(Σ1 ) and ∆′2 ⊂ ∆(Σ2 ) decompose the spherical system (S p , Σ, A). Proof. By definition, the subsets ∆′1 and ∆′2 are distinguished. (i) They are different from the empty set as required and no colour can be in the intersection of ∆(Σ1 ) and ∆(Σ2 ) thanks to lemma 2.4. (ii) The subsets ∆′1 and ∆′2 satisfy the ∗-property. Their union also satisfies the ∗-property since Ξ Ξ Ξ = ′ ∩ ′ ∆′1 ∪ ∆′2 ∆1 ∆2 and

Σ1 Σ2 Σ = Σ \ (Σ ∪ Σ ) ∪ ′ ∪ ′. 1 2 ∆′1 ∪ ∆′2 ∆1 ∆2 (iii) The subsets Σ1 and Σ2 are disjoint. (iv) Let us suppose that α ∈ supp(Σ1 ) and β ∈ supp(Σ2 ) with α nonorthogonal to β: analyzing all the cases for α, β of type a, a′ , b, p we get that neither ∆(α) can be included in ∆(Σ1 ) nor ∆(β) can be included in ∆(Σ2 ). (v) By the hypothesis, ∆′1 or ∆′2 is smooth. A localization (S p ∩ supp(Σ′ ), Σ′ , A(S ∩ Σ′ )) of a strongly ∆-connected component is said to be isolated if the partition supp(Σ′ ) ⊔ (supp(Σ) \ supp(Σ′ )) gives a factorization of the localization in supp(Σ) of the spherical system (S p , Σ, A). An isolated localization is a fortiori erasable. Reduction. The aim of this paragraph is to list all primitive spherical systems up to an automorphism of the Dynkin diagram of G. The list of the primitive spherical systems of type A can be taken from [L3] and it is included in Table 3; we have added two further cases which were missing in [L3]. 2There exists a weaker property called ∆-connectedness also introduced by Luna in [L3], but its generalization to type A D doesn’t seem to be as useful as in type A. 3This requirement is slightly weaker than that found in [L3].

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The first step to obtain the complete list in type A D consists in finding all the cuspidal strongly ∆-connected spherical systems. One can proceed by taking a spherical root, attaching to it any other spherical root to obtain a strongly ∆connected pair and going on recursively by attaching a new spherical root anywhere it is possible. We obtain the following, where the labels are those of Table 3. Starting from a spherical root of type dm : ac(n) n ≥ 3 odd, d(n) n ≥ 4 with only one spherical root of type dn (ac(3) will be denoted also by d(3)), do(p + q) q ≥ 3, dc(n) n even, dc′ (n) n even, dc(n) n odd, dc♯ (n) (the case D1(i) of Table 2). Starting from a root of type a1 × a1 without roots of type dm : aa(p + q + p), aa(p, p), aa∗ (p + 1 + p), do(p + 2), dd(p, p) p ≥ 4. Starting from a root of type a′1 without roots of type dm and a1 × a1 : ao(n), do(n). Starting from a root of type am , m ≥ 2, with only roots of type am : aa(n) n ≥ 2 with only one spherical root of type aq , ac∗ (n) n ≥ 3, dc∗ (n), ds(n), ds∗ (4). Now we must find all cuspidal strongly ∆-connected spherical systems with Σ containing only spherical roots of type a1 . In this case we have S p = ∅ and Σ = S. Lemma 3.10. If such a spherical system has a projective element δ, then hρ(δ), αi = 1 for all α ∈ Σ, namely: \ δ∈ A(α). α∈Σ

Proof. Two spherical roots α and β of type a1 are strongly ∆-adjacent if and only if there is a colour δ in A(α) ∩ A(β), namely we have the cases A7(i) and A8 of Table 2. If α and β are spherical roots of a spherical system with the above properties, this colour δ is the unique element of A that can be projective. We will call n-comb a spherical system of rank n satisfying the hypotheses of the above lemma. In order to obtain all cuspidal strongly ∆-connected spherical systems with only spherical roots of type a1 without projective elements, one can proceed analogously as above. The list is the following: aa(1) with only one spherical root of type a1 , ax(1, 1, 1), ay(p, p) p ≥ 2, ay(p, (p − 1)) p ≥ 2, ay(n) n ≥ 4 even, az(3, 3), az(3, 2), az(3, 1), ae6 (6), ae6 (5), ae7 (7), ae7 (6), ae7 (5), dy(p, p) p ≥ 4, dy(p, (p − 1)) p ≥ 4, dy(p, (p− 2) p ≥ 4, dz(4, 4), dz(4, 3), dz(4, 2), dz(4, 1), de6 (7), de6 (6), dy(7), de7 (8), de7 (7), de8 (8), de8 (7). The next step will consist in gluing two or more strongly ∆-connected components together. The following consideration about the properties of being erasable or isolated will be useful to simplify this procedure. Let (S p , Σ, A) be a cuspidal spherical system and let Σ′ give a strongly ∆connected component (S p ∩ supp(Σ′ ), Σ′ , A(S ∩ Σ′ )). There are only the following two situations where an element δ ∈ A may correspond to a colour not belonging to ∆(Σ′ ): - The colour δ is said to be external if there is a unique simple root α with δ ∈ A(α) and it is not in ∆(Σ′ ) since it takes a negative value on a spherical root γ outside Σ′ . Then the root α is nonorthogonal to γ. - The colour δ is said to be a bridge if there are more than one simple root, let us say αi for i = 1, . . . , r, such that δ is in A(αi ), and it takes a negative value on a spherical root γ outside Σ′ . Then all the roots αi are


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nonorthogonal to γ, which is necessarily of type am , and r is equal to 2 or 3. In most cases a look at the set of colours of a strongly ∆-connected component easily shows that it is necessarily isolated, erasable or quasi-erasable. The strongly ∆-connected component (S p ∩ supp(Σ′ ), Σ′ , A(S ∩ Σ′ )) is isolated if it is isomorphic to one of these: do(p + q), dc(n) for n ≥ 6 even, dc′ (n), dc(n) for n odd, aa(p + q + p) for p ≥ 2, aa(p, p) for p ≥ 2, aa∗ (p + 1 + p), do(p + 2), dd(p, p), ao(n), do(n), ay(n) for n ≥ 6, dy(p, p) for p ≥ 5, de6 (6), dy(7), de7 (7), de8 (7). If the strongly ∆-connected component (S p ∩ supp(Σ′ ), Σ′ , A(S ∩ Σ′ )) is isomorphic to ac(n) for n ≥ 5, aa(1 + q + 1) for q ≥ 2, ay(4) or ae7 (5) then either it is isolated or supp(Σ′ ) is nonorthogonal to the support of an n-comb and (S p , Σ, A) has a projective element. The strongly ∆-connected component (S p ∩ supp(Σ′ ), Σ′ , A(S ∩ Σ′ )) is erasable if it is isomorphic to one of these: dc♯ (n), aa(1 + 1 + 1), n-comb for n ≥ 4, az(3, 3), ae6 (6), ae6 (5), ae7 (7), ae7 (6), dy(4, 4), dz(4, 4), dz(4, 3), dz(4, 2), dz(4, 1), de6 (7), de7 (8), de8 (8). The strongly ∆-connected component (S p ∩ supp(Σ′ ), Σ′ , A(S ∩ Σ′ )) is quasierasable if it is isomorphic to one of these: dc∗ (n), ds(n), ds∗ (4). We examine the remaining strongly ∆-connected components case by case: - d(n) for n ≥ 3. If it is not isolated, it is neither erasable nor quasi-erasable. - aa(1, 1). Analogously, if it is not isolated, it is neither erasable nor quasierasable. The strongly ∆-connected component aa(1, 1) will be denoted also by d(2), since it can be considered as limit case of d(n) for n > 2. - aa(n) for n ≥ 2. It is erasable only if it lies at one end of a connected component (in the usual sense) of the Dynkin diagram. - aa(1). If it has no projective elements then the functionals of its two colours take a negative value respectively on two distinct spherical roots. Hence, it does not lie at the end of a connected component of the Dynkin diagram or the spherical system admits the following erasable localizations: aa(1) + ac∗ (n), aa(1) + ac♭ (n), aa(1) + ac♭♭ (4) (see Table 3). - ac∗ (n). In case its support contains a “bifurcation” of the Dynkin diagram, i.e. a simple root nonorthogonal to three simple roots, and there are no projective elements, we have necessarily one of the following two erasable localizations: ac∗ (n) + aa(1), ac∗ (3) + 2−comb (see Table 3). In case it does not lie on a bifurcation, it is erasable if at the end of a connected component of the Dynkin diagram or if n is odd, it is quasi-erasable if n is even. - 3-comb. It has a projective element or its support is made of three mutually orthogonal simple roots joined by a bridge. In the latter case it is part of one of the following erasable localizations: 3−comb+ 2−comb, 3−comb+ aa(n) (see Table 3). - 2-comb. Analogously, it has a projective element or its support is made of two orthogonal simple roots joined by a bridge. - ax(1, 1, 1). If there are no projective elements then either it lies at the end of a connected component of the Dynkin diagram and it is quasi-erasable, either it is part of ax(1 + p + 1, 1) that is quasi-erasable or it is part of af (5) that is isolated.

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- ay(p, p). There are different colours that may be a bridge. If there is no bridge then it is erasable. Otherwise it is part of one of the following erasable localizations: ay(p + q + p), dy ∗ (3 + q + 3), ay(2, 2) + 2−comb (see Table 3). - ay(p, p − 1). If there is no bridge then it is quasi-erasable, or erasable if at the end of a Dynkin diagram. Otherwise it is part of one of the following localizations: ay(p + q + (p − 1)) erasable, dy ∗ (3 + q + 2) isolated, ay(2, 1) + 2−comb isolated (see Table 3). - az(3, 2). If there is no bridge then it is erasable. Otherwise it is part of one of the following localizations: dy(3 + q + 2) isolated, dz(3 + q + 2) erasable (see Table 3). - az(3, 1). There is only one colour that may be a bridge and in this case we can have only dz(3+q +1) that is isolated, if this colour is not a bridge then az(3, 1) is quasi-erasable or even erasable if it does not lie on a bifurcation. - dy(p, p − 1). There is only one colour that may be a bridge and in this case we can have only dy(p + q + (p − 1)) that is isolated, if this colour is not a bridge the strongly ∆-connected component itself is erasable. - dy(p, p − 2). There is only one colour that may be a bridge and in this case we can have only dy ′ (p + q + (p − 2)) that is isolated, if this colour is not a bridge the strongly ∆-connected component itself is quasi-erasable. We are ready to complete the list of primitive spherical systems: we have to “put together” the components obtained above in all possible ways such that the obtained system turns out to be indecomposable, without projective colours. Thanks to Lemma 3.9 and subsequent considerations, the procedure is quite straightforward and leads to the list of Table 3. Example. Let us start with az(3, 3): first of all, it is a primitive system itself. We go on adding other components, taking into account the fact that az(3, 3) is erasable whenever it is a part of a larger system. Therefore we may suppose that there is no other strongly ∆-connected component (more precisely, no union of such components) being quasi-erasable. The list at the end of the preceding paragraph gives us the other possible components: d(n), aa(n), 2-comb. A copy of d(n) or aa(n) can be added to az(3, 3), obtaining the systems: az ∼ (3 + q + 3), az(3, 3) + d(n). An additional copy of d(n) gives the systems az ∼ (3 + q + 3) + d(n) and az(3, 3) + d(n1 ) + d(n2 ). All other possibilities (including those containing a 2-comb) bring some union of the added components being quasi-erasable. 4. The proof of the primitive cases In this section we complete the proof of Theorem 2.9, checking that every primitive spherical system corresponds to a wonderful variety, unique up to Gisomorphism. We divide the proof for each case in two parts. In the “uniqueness” part we suppose that there exists a wonderful subgroup H such that the wonderful embedding of G/H has the required spherical system, and we prove that H is uniquely determined up to conjugation. This will provide a candidate H and in the “existence” part we prove that it is wonderful and that it actually corresponds to our spherical system.


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For all cases where G is of type A this has already been proven in [L3], thus we suppose G be semisimple adjoint of mixed type A D, not of type A. Wherever it is convenient, we will work on a covering of G rather than on G itself. The next two technical lemmas will be widely used through all proofs. Lemma 4.1 ([L3]). Let H be a wonderful subgroup of G and let (S p , Σ, A) be its spherical system. Then the following identities hold: (i) dim G/PS p + rank Σ = dim G/H, (ii) rank Ξ(H) = rank ∆ − rank Σ. Lemma 4.2. Let (S p , Σ, A) be a spherical system of type A D. (i) In general, rank Σ ≤ rank S and dim G/PS p = dim PSup ≤ dim B u . (ii) If Σ contains a spherical root of type dm or a1 × a1 then rank Σ < rank S. (iii) If Σ contains a spherical root of type am , for any m ≥ 1, then rank Σ < rank ∆ or (S p , Σ, A) can be localized to ax(1, 1, 1) or ds∗ (4). (iv) Every strongly ∆-connected component with only spherical roots of type a1 different from ax(1, 1, 1) and ds∗ (4) is such that rank ∆ − rank Σ = 1. Proof. All the statements are quite obvious; in order to prove iii in case of m = 1 and iv it is sufficient to check them for all cuspidal strongly ∆-connected spherical systems with only spherical roots of type a1 . 4.1. Reductive cases. Let H be a spherical subgroup of G. Then the homogeneous space G/H is affine, or equivalently the subgroup H is reductive, if and only if there exists a linear combination X ξ= cγ γ γ∈ΣG/H

with cγ ≥ 0 such that hρ(δ), ξi > 0 for all δ ∈ ∆G/H . The primitive systems which satisfy this property are: do(p + q), dc(n) for n even, dc′ (n), dc(n) for n odd, do(p + 2), dd(p, p), do(n), ds∗ (4) and aa(1) + ds∗ (4). On the other hand, reductive spherical subgroups are classified ([Kr], [B1]); we report in Table 5 the list of the indecomposable connected reductive spherical subgroups. Table 5: Indecomposable connected reductive spherical subgroups of semisimple groups. H

G

SO(n) S(GL(n) × GL(m)) SL(n) × SL(m) Sp(2n) Sp(2n) GL(1) · Sp(2n) GL(n) SL(n) GL(n) SO(n) × SO(m) Spin(7) Spin(7) G2

SL(n) SL(n + m) SL(n + m) SL(2n) SL(2n + 1) SL(2n + 1) SO(2n) SO(2n) SO(2n + 1) SO(n + m) SO(8) SO(9) SO(7)

n≥2 n≥m≥1 n>m≥1 n≥2 n≥1 n≥1 n≥2 n ≥ 3 odd n≥2 n ≥ m ≥ 1 and n + m ≥ 3

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Table 5: Indecomposable connected reductive spherical subgroups of semisimple groups (continuation). H

G

G2 SL(2) · Sp(4) SO(2) × Spin(7) GL(n) Sp(2n) × Sp(2m) GL(1) × Sp(2n − 2) A2 A1 × A1 B4 C3 × A 1 C4 F4 D5 GL(1) · D5 A5 × A1 GL(1) · E6 A7 D6 × A1 D8 A1 × E 7 H

SO(8) SO(8) SO(10) Sp(2n) Sp(2n + 2m) Sp(2n) G2 G2 F4 F4 E6 E6 E6 E6 E6 E7 E7 E7 E8 E8 H ×H

SL(2) SL(2) × Sp(2n) Sp(4) × Sp(2n) SO(n) Spin(7) SL(2) × Sp(2m)× ×Sp(2n) SL(2) × SL(2)× ×Sp(2m) × Sp(2n) SL(2) × Sp(2l)× ×Sp(2m) × Sp(2n) SL(n) × GL(1) SL(2) × SL(m)× ×Sp(2n) SL(2) × SL(m)× ×Sp(2n) × GL(1)

SL(2) × SL(2) × SL(2) SL(2) × Sp(2n + 2) Sp(4) × Sp(2n + 4) SO(n) × SO(n + 1) Spin(7) × SO(8) Sp(2m + 2) × Sp(2n + 2)

H simple group embedded diagonally in H × H embedded diagonally (u, v) 7→ (u, u ⊕ v) (u, v) 7→ (u, u ⊕ v) u 7→ (u, i(u)) u 7→ (u, i(u)) (t, u, v) 7→ (t ⊕ u, t ⊕ v)

Sp(4) × Sp(2m + 2)× ×Sp(2n + 2) Sp(2l + 2) × Sp(2m + 2)× ×Sp(2n + 2) SL(n) × SL(n + 1) SL(m + 2) × Sp(2n + 2)

(t, u, v, w) 7→ 7→ (t ⊕ u, t ⊕ v, u ⊕ w) (t, u, v, w) 7→ 7→ (t ⊕ u, t ⊕ v, t ⊕ w) (u, λ) 7→ (u, λu ⊕ λ−n ) (u, v, w) 7→ (u ⊕ v, u ⊕ w)

SL(m + 2) × Sp(2n + 2)

(u, v, w, λ) 7→ 7→ (λm u ⊕ λ−2 v, u ⊕ w)

n≥1 n≥m≥1 n≥1

In particular, for G of type D we have the following indecomposable connected reductive spherical subgroups. - GL(n) ⊂ SO(2n), n ≥ 2, dim G/H = n(n − 1), rank Ξ(H) = 1. - SL(n) ⊂ SO(2n), n ≥ 3 odd, [N (H) : H] infinite. - SO(n)×SO(m) ⊂ SO(n+m), n ≥ m ≥ 1, n+m ≥ 4 even, dim G/H = nm, rank Ξ(H) equal to 0 if m, n 6= 2, 1 if m = 2 or n = 2, 2 if m = n = 2. - Spin(7) ⊂ SO(8), dim G/H = 7, rank Ξ(H) = 0. - G2 ⊂ SO(8), dim G/H = 14, rank Ξ(H) = 0.


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- SL(2) · Sp(4) ⊂ SO(8), dim G/H = 15, rank Ξ(H) = 0. - SO(2) × Spin(7) ⊂ SO(10), dim G/H = 23, rank Ξ(H) = 1. - SO(2n) ⊂ SO(2n) × SO(2n) embedded diagonally, n ≥ 2, dim G/H = n(2n − 1), rank Ξ(H) = 0. This table provides a candidate H for each of the above primitive systems, or at least a candidate for the connected component H ◦ of H containing the identity. In the latter case we have to guess H, knowing that H ◦ ⊂ H ⊂ NG (H) = NG (H ◦ ), where the last equality is known from the theory of spherical varieties. In general, the normalizer NG (H) of a wonderful subgroup H acts on the colours of G/H and the set of the colours of G/NG (H) is a quotient of the set of the colours of G/H such that the colours permuted by NG (H) are identified. The normalizer NG (H) can permute only the colours having the same associated functional, and we have that the only case where two colours can have the same functional is the case of two colours δα+ and δα− moved by the same simple root α being also a spherical root. This fact can be easily deduced from the list of the rank two wonderful varieties, and in type A D it holds that the spherical system associated NG (H), if it is different from the spherical system associated to H, can be obtained from the latter by substituting some spherical roots α ∈ S ∩ Σ with the spherical roots 2α (see also Proposition 5.1). Now, once we have a candidate H, ad-hoc considerations are needed in order to prove that H satisfy our requirements, with the help of lemmas 4.1 and 4.2 and the stringent restrictions imposed by the axioms of spherical systems. do(p + q) (p ≥ 1, q ≥ 3). To see that H must be reductive here we can take ξ = c1 2α1 + . . . + cp 2αp + cp+1 γp+1 , where Σ = {2α1 , . . . , 2αp , γp+1 }, with ci = f (i) for any concave function of the interval [1, p + 1] with 2f (1) > f (2) and f (p) < f (p + 1), for example ci = i + 1 − (1/(p + 2 − i)). For the next cases we will omit this easy check that H is reductive. Uniqueness. We have here G = SO(2p + 2q). The spherical system is such that dim G/PS p = dim PSup = (p + 1)(2q + p − 2), rank Σ = p + 1 and rank ∆ = p + 1. Hence by Lemma 4.1 we have dim G/H = (p + 1)(2q + p − 1) and rank Ξ(H) = 0. Looking at Table 5 we see that these conditions force the connected component H ◦ of the identity in H to be conjugated to SO(p + 1) × SO(2q + p − 1), or GL(p + q) if p+q = (q−1)2 . Let us prove that the latter choice is impossible. If p+q is even then [NG (GL(p + q) : GL(p + q)] = 2. Now H cannot be isomorphic to GL(p + q) since rank Ξ(GL(p + q)) = 1. The only other chance would be H = NG (GL(p + q) up to conjugation, but the existence part of the proof for dc′ (n) (page 39) shows that NG (GL(p+ q)) is wonderful and the associated spherical system is of type dc′ (n). If p+q is odd NG (GL(p+q)) = GL(p+q) and rank Ξ(GL(p+q)) = 1. Therefore, H ◦ is conjugated to SO(p+1)×SO(2q+p−1). The normalizer is S(O(p+1)×O(2q+p−1)) and [N (H ◦ ) : H ◦ ] = 2, hence H can only be H ◦ or N (H ◦ ). Now, we will see in the existence part of the proof that N (H ◦ ) is wonderful and the associated spherical system is actually do(p + q). By the results on spherically closed subgroups in [L3] the lattice ΞG/H ◦ contains properly ΞG/H , hence H ◦ and N (H ◦ ) cannot have the same spherical system. This forces H = N (H ◦ ). Existence. Let us put H = S(O(p + 1) × O(2q + p − 1)). Since rank Ξ(H) = 0, we have rank Σ = rank ∆, hence the associated spherical system is cuspidal and, by Lemma 4.2, the set Σ does not contain roots of type am for m ≥ 1. Indeed, the

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case of a component of type ds∗ (4) can be excluded directly by noticing that there is a spherical root of type a3 that can be nonorthogonal only to a spherical root of type am (see Table 2). Moreover, dim PSup + rank Σ = dim G/H = (p + q)(p + q − 1) − q(q − 2). In the case of S p = ∅, we would have rank Σ = p + q − q(q − 2) since PSup = B u and no roots of type dm , but this is impossible. Hence, we have that S p 6= ∅ and Σ contains at least one root of type dm . If this root does not lie at the end of the diagram like in do(p + q), we get dc′ (p + q), since a spherical root of type d3 on a Dynkin diagram of type A can be nonorthogonal only to a root of the same type. The case dc′ (p + q) can be excluded since it is not invariant under the exchange of the simple roots αp+q−1 and αp+q , while if consider the corresponding outer involutive automorphism σ we have σ(H) = H. Hence, there is only one root of type dm , it lies at the end of the diagram and the unique possible spherical system is do(p + q). do(p + 2). Suppose at first n = p + 2 > 4. Uniqueness. This case is similar to do(p + q), here we can take p = n − 2 and q = 2. The subgroup H must be reductive. We have dim G/H = (n − 1)(n + 1) and rank Ξ(H) = 0. If G = SO(2n), by Table 5 we get that H ◦ ∼ = SO(n − 1) × SO(n + 1), and analogously to do(p + q) we get that H = NG (H ◦ ) = S(O(n − 1) × O(n + 1)). Existence. H is wonderful. The identities dim PSup + rank Σ = (n − 1)(n + 1) and rank ∆ − rank Σ = 0 force rank Σ ≥ n − 1. In general rank Σ ≤ n. If rank Σ = n then rank S p = 1, but no root of type am can belong to Σ, hence rank Σ = n − 1 and S p = ∅. We obtain the spherical system do(p + 2). Let n = 4. Notice that in this case we have three different spherical systems of rank 3. q qe e q e q q q q q q e e e e e e @ @ @ @e q @e q @q e We have dim G/H = 15 and rank Ξ(H) = 0. Our G is SO(8), and by Table 5 we get that H ◦ can be isomorphic only to SO(3) × SO(5) or SL(2) · Sp(4). In the first case we have that NSO(8) (SO(3) × SO(5)) = (SO(3) × SO(5)) · CSO(8) as above. In the second case we have that SL(2) · Sp(4) ∼ = Spin(3) · Spin(5) ⊂ Spin(7) ⊂ SO(8) is equal to its normalizer. Let us consider the outer involutive automorphism σ of SO(8) that exchanges the simple roots α3 and α4 : we have that σ(S(O(3) × O(5))) = S(O(3) × O(5)) while σ(SL(2) · Sp(4)) is isomorphic to SL(2) · Sp(4) but they are not conjugated. Therefore, we have three possible subgroups up to conjugation: S(O(3) × O(5)), SL(2) · Sp(4) and σ(SL(2) · Sp(4)). They are spherical and hence wonderful, and their spherical systems are all of type do(p + 2) (one proceeds like in the case n ≥ 4). The first subgroup has the first spherical system of the above figure, since it is the unique one fixed by σ. The other subgroups are analogously in correspondence with the remaining two cases. do(n). Uniqueness. We have dim G/H = n2 and rank Ξ(H) = 0. If G = SO(2n), by Table 5 we get H ◦ ∼ = SO(n) × SO(n), and [NG (H ◦ ), H ◦ ] = 4. The ◦ subgroup NG (H ) is wonderful and, thanks to the existence part, the spherical system associated to it is do(n); we conclude that H must be NG (H ◦ ) as in case do(p + q). Existence. From the identities dim PSup + rank Σ = n2 and rank ∆ − rank Σ = 0 we get S p = ∅, rank Σ = n and rank ∆ = n. Therefore, we get the spherical system do(n).


39

dc(n) (n ≥ 4 even). Uniqueness. By Lemma 4.1 we have dim G/H = (n(n − 1) − n/2) + n/2 = n(n − 1) and rank Ξ(H) = 1. Our G is SO(2n), from Table 5 we see that if n > 4 then H ◦ ∼ = GL(n) and H = H ◦ , since rank Ξ(NG (GL(n))) = 0. Existence. The spherical subgroup H has index 2 in its normalizer NG (H). Let us consider the functions f1 , f2 ∈ C[SO(2n)] defined by the minors (n + 1 . . . 2n | 1 . . . n) and (n + 1 . . . 2n | n + 1 . . . 2n), respectively. Since they are (B × H)-proper and they are exchanged by the right action of NG (H), there are two colours of G/H exchanged by N (H). Therefore H is spherically closed and hence wonderful ([L3],[Kn]). We have dim PSup + rank Σ = n(n − 1) and rank ∆ − rank Σ = 1. Hence, S p is non-empty and there exists at least a spherical root of type dm , since by the above identities any root of type am with m ≥ 3 cannot be in Σ. It remains only the spherical system dc(n). dc′ (n) (n ≥ 4 even). Uniqueness. Analogously, dim G/H = n(n − 1) and rank Ξ(H) = 0. Here G = SO(2n); by Table 5 and, by the existence part of the proof for the case do(p + q), we get H ∼ = NG (GL(n)). Existence. The subgroup H is wonderful, since it is equal to its normalizer. Moreover, we already know the spherical system of H ◦ and hence we get the case dc′ (n). dc(n) (n ≥ 5 odd). Uniqueness. We have dim G/H = n(n − 1) and rank Ξ(H) = 1. Here G = SO(2n); by Table 5 we get H ◦ ∼ = GL(n). Moreover, if n is odd then GL(n) = NG (GL(n)) , hence H = H ◦ . Existence. H is wonderful. We have dim PSup + rank Σ = n(n − 1) and rank ∆ − rank Σ = 1. Hence, S p 6= ∅ and there exists at least a spherical root of type am with m ≥ 3, since we cannot obtain any spherical system satisfying the above identities with only roots of type dm . There are no other choices than the spherical system dc(n). dd(p, p) (p ≥ 4). Uniqueness. We have dim G/H = p(2p − 1). We have G = SO(2p) × SO(2p), by Table 5 we get that H ◦ is equal to SO(2p) embedded diagonally in G. The normalizer NG (H ◦ ) of H ◦ is CG · H ◦ , therefore H is NG (SO(2p)). Existence. From rank ∆ = rank Σ follows that the spherical system of H is cuspidal and without any root of type am . Since H is not a direct product, there exists a root of type a1 × a1 whose support is not included in one single connected components of the Dynkin diagram. The spherical system is forced to be dd(p, p). ds∗ (4). Uniqueness. We have dim G/H = 14 and rank Ξ(H) = 0. Our G is SO(8), by Table 5 we get H ◦ ∼ = G2 . Moreover, we have NSO(8) (G2 ) = G2 · CSO(8) . Existence. By the identities dim PSup + rank Σ = dim B u + 2 and rank ∆ = rank Σ we get rank S p = 1 and hence at least one root of type a3 . Therefore, as seen in Lemma 4.2, we get the spherical system ds∗ (4). aa(1) + ds∗ (4). Uniqueness. We have dim G/H = 15 and rank Ξ(H) = 1. If G = SO(10), by Table 5 we get H ◦ ∼ = SO(2) × Spin(7). Moreover, we have NSO(10) (H ◦ ) = H ◦ . Existence. By the identities dim PSup + rank Σ = dim B u + 3 and rank ∆ = rank Σ + 1 we get that rank S p = 1 and there are roots of type am . In particular, the only possibility is to take three roots of type a3 and one root of type a1 . Hence, we must attach to ds∗ (4) a root of type a1 . There are two possibilities: in one case H would be included, by Proposition 3.2, in the parabolic induction of G2 , but this is false. We get the spherical system aa(1) + ds∗ (4).

40


4.2. Nonreductive cases. As above, for each case we prove uniqueness and existence of the associated wonderful subgroup H. For the uniqueness part we use the following strategy, with some marginal changes. It is the typical argument used by Luna in [L3]. Let (S p , Σ, A) be a primitive spherical system for G and let us suppose that there exists a subgroup H of G such that the wonderful embedding of G/H exists and has (S p , Σ, A) as its spherical system. First step. We find a distinguished subset ∆′ of ∆ with the ∗-property such that (S p , Σ, A)/∆′ is a parabolic induction of a reductive case and there exists a minimal parabolic distinguished subset ∆′′ such that it contains ∆′ . The quotient (S p , Σ, A)/∆′ corresponds to a unique wonderful variety X1 thanks to the previous paragraph, and we have G-morphisms: Φ

Ψ

′

∆ X −→ X1 −→ G/P

where Ψ◦Φ∆′ = Φ∆′′ and P is the parabolic subgroup containing B − and associated to the subset of simple roots S p /∆′′ . Let H1 ⊂ G be a subgroup such that X1 is the wonderful embedding of G/H1 : we may suppose that H ⊂ H1 ⊂ P . P is a minimal parabolic subgroup containing H thanks to the properties of ∆′′ . We have also the inclusion H u ⊂ P u between the unipotent radicals: indeed, the morphism: H ֒→ P −→ P/P u yields the inclusion: H/(H ∩ P u ) ֒→ P/P u P/P u is reductive and its subgroup H/(H ∩ P u ) is not included in any proper parabolic subgroup, hence it is reductive and this implies H u ⊂ P u . Therefore we may suppose to have two Levi decompositions H = L · H u and P = L(P ) · P u , where L ⊂ L(P ) and H u ⊂ P u ; this also implies that L is spherical in L(P ) and not contained in any proper parabolic subgroup. Moreover, we may suppose to have a Levi decomposition H1 = L1 · H1u , where L ⊂ L1 . The commutator subgroup (L, L) is a reductive spherical subgroup of (L1 , L1 ). Let Q be the minimal parabolic subgroup containing B − such that Qr ⊂ H1 ⊂ Q: in such a case we have Qu = H1u . In many cases Q is equal to P . Second step. We show that (L, L) is forced to be equal to (L1 , L1 ), so that we also have H u ⊂ H1u . This is proven checking all possible candidates for (L, L) provided by the list of reductive spherical subgroups in Table 5. For each such candidate, we consider H ′ = P r · (L, L), which will be always wonderful. The inclusion H ⊂ H ′ would induce a G-morphism Φ with connected fibers from X onto the wonderful embedding of G/H ′ , whose spherical system is known thanks to the previous paragraph. But now the associated distinguished subset of colours ∆Φ will not exist, unless we had chosen (L, L) = (L1 , L1 ). Therefore we are in the following situation: H

=

H1

=

C ∩ C1

·

(L, L) k · (L1 , L1 )

· Hu ∩ · H1u

where C and C1 are the connected centers of H and H1 , respectively. We have that the Lie algebra of H u is an (L, L)-submodule of the Lie algebra of H1u . Moreover,


41

since we know C1 and dim C, we can compute the codimension of H u in H1u using Lemma 4.1. Third step. We analyze the decomposition of the Lie algebra of H1u into simple L1 -modules: ad-hoc considerations about the spherical system arise conditions on C and the Lie algebra of H u in such a way that H is uniquely determined up to conjugation. For the existence part of the proof, namely once we have a candidate H, we prove it is spherical and then we use the same kind of indirect arguments of those in the previous section. To prove that H is spherical we use the following: Lemma 4.3 ([B1]). Let Q and L(Q) be the above defined subgroups of G, let n1 and n denote the Lie algebras of H1u and H u , respectively. If there exists a Borel subgroup B ′ of L(Q) such that B ′ L is open in L(Q) and B ′ ∩ L has an open orbit in n1 /n, then H is spherical. Moreover, notice that the proof of the existence for a primitive spherical system implies the existence for everyone of its localizations. Therefore, for any localized case it is necessary to provide only the uniqueness part of the proof. Strongly ∆-connected cases. In the following strongly ∆-connected cases we get always that P = Q. dc∗ (n). We have S p = ∅, Σ = {α1 +α2 , . . . , αn−3 +αn−2 , αn−2 +αn−1 , αn−2 + αn } and A = ∅. The set of colours is ∆ = {δα1 , . . . , δαn }. Uniqueness. Suppose n odd. The set ∆′′ = {δα1 , δα2 , . . . , δαn−2 } is the minimum parabolic subset of colours, and it is associated to the parabolic subgroup PS p /∆′′ where S p /∆′′ = S \{αn−1 , αn }. The Luna diagram of the quotient spherical system (S p , Σ, A)/∆′′ is the following. qe q q q q q @ q @e Consider the distinguished set of colours ∆′ = {δα1 , . . . , δα2i+1 , . . . , δαn−2 } ⊂ ∆′′ : it is such that S p /∆′ = {α1 , . . . , α2i+1 , . . . , αn−2 } and Σ(∆′ ) = {αn−2 + αn−1 , αn−2 + αn }. We have Ξ/∆′ = {ξ : hρ(δ), ξi = 0 ∀δ ∈ ∆′ and hγ ∗ , ξi = 0 ∀γ ∈ Σ(∆′ )}, so the spherical roots of the quotient are the indecomposable elements of the semigroup n−3 X ci (αi + αi+1 ) : ci ≥ 0 ∀i and ci − ci+1 = 0 ∀i odd}, { i=1

i. e. Σ/∆′ = {α1 + 2α2 + α3 , . . . , α2i−1 + 2α2i + α2i+1 , . . . , αn−4 + 2αn−3 + αn−2 }. The quotient spherical system (S p , Σ, A)/∆′ is a parabolic induction of ac(n − 2). qe ppppppppppqe q q pppp q ppppppppppppppqe q ppppppppppppppqe q @ @e q Our G is SO(2n). Let Q denote the parabolic subgroup PS−p /∆′′ , we have Q = Qu L(Q), where L(Q) ∼ = GL(n−1)×GL(1). Let H1 denote the wonderful subgroup associated to the spherical system (S p , Σ, A)/∆′ . It is equal to H1u L1 , where H1u = Qu , (L1 , L1 ) ∼ = Sp(n − 1) and C1 is the connected center of L(Q), where dim C1 = 2 = rank(∆ \ ∆′ ) − rank Σ/∆′ . Indeed, the wonderful subgroup of SL(n − 1)

42


associated to the spherical system ac(n − 2) is isomorphic to Sp(n − 1) · CSL(n−1) . So (L1 , L1 ) is the subgroup of the matrices 

A  0   0 0

0 1 0 0

0 0 1 0

 0  0   0 J tA−1 J

where A ∈ Sp(n − 1), and C1 is the subgroup of the matrices    

c1 I 0 0 0

0 c2 0 0

0 0 c−1 2 0

0 0 0 c−1 1 I

   

where c1 , c2 ∈ GL(1). The Lie algebra n1 ⊂ so(2n) of H1u is the subalgebra of the matrices skew symmetric with respect to the skew diagonal 

0  M1   M2 M3

0 0 0 −J tM2

0 0 0 −J tM1

 0 0   0  0

where M1 and M2 are row (n − 1)-vectors and M3 is a square (n − 1)-matrix skew symmetric with respect to the skew diagonal. Let H = H u L be a wonderful subgroup with spherical system dc∗ (n). We can suppose H ⊂ H1 and L ⊂ L1 . By Lemma 4.1 we have that dim H1 − dim H = n and dim C = 1. The subgroup (L, L) of (L1 , L1 ) is spherical; it is also reductive because it is contained in no proper parabolic subgroup (thanks to the minimality of ∆′′ ). By Table 5, (L, L) can be isomorphic only to Sp(n − 1) and hence it is equal to (L1 , L1 ). In particular, dim H1u − dim H u = n − 1. Call mi the subspace of n1 given by the condition Mj = 0 for all j 6= i. As a (L, L)module, n1 has three isotypic components: the first component is m1 +m2 where m1 and m2 are simple L1 -submodules of dimension n − 1 (two copies of the dual of the standard representation Cn−1 ), the second component is a simple L1 -submodule of dimension ((n − 1)(n − 2)/2) − 1 and the third component is a simple L1 -submodule V2 of dimension 1. The sum of these two last isotypic components is m3 ∼ = ( Cn−1 )∗ . The Lie algebra n ⊂ n1 of H u is an L-submodule of codimension n − 1. Being a Lie subalgebra, it must be the sum of m3 and a submodule m ⊂ m1 + m2 of codimension n − 1. The subgroup C1 acts as follows, for all c ∈ C1 and for all Mi ∈ mi : −1 −1 c.M1 = c2 c−1 1 M1 , c.M2 = c2 c1 M2 . Moreover, H must be wonderful, hence −1 −1 ◦ of finite index in its normalizer. Therefore, C = {c2 c−1 1 = c2 c1 } = {c2 = 1} and m is diagonal in m1 + m2 , namely defined by an equation µ1 M1 + µ2 M2 = 0 with µi 6= 0. This determines C and H u up to conjugation. Since H contains the center CG of the group G, the Levi subgroup L is isomorphic to CG · C · Sp(n − 1). Moreover H = NG (H), hence it is uniquely determined up to conjugation. Now suppose n even. The subset ∆′′ = {δα2 , δα3 , . . . , δαn−2 } is a minimum parabolic subset and S p /∆′′ = S\{α1 , αn−1 , αn }. The set ∆′ = {δα2 , . . . , δα2i , . . . , δαn−2 } is distinguished, with S p /∆′ = {α2 , . . . , α2i , . . . , αn−2 } and Σ(∆′ ) = {α1 +α2 , αn−2 + αn−1 , αn−2 + αn }, so that Σ/∆′ = {α2 + 2α3 + α4 , . . . , α2i + 2α2i+1 + α2i+2 , . . . , αn−4 + 2αn−3 + αn−2 }. If n > 4 the quotient spherical system is a parabolic induction of ac(n − 3), if n = 4 it has rank zero. qe q q ppppppppppppppqe q ppppppppppppppqe q e q ppppppppppppppqe q @ @e q


43

Again, G = SO(2n). Set Q = PS−p /∆′′ , we have L(Q) ∼ = GL(1)×GL(n−2)×GL(1). Here H1 is such that (L1 , L1 ) ∼ Sp(n − 2) and C is the connected center of L(Q), = 1 where dim C1 = 3. Let I and J denote the diagonal and skew diagonal unitary (n − 2) × (n − 2) matrices; we have that (L1 , L1 ) is the subgroup of the matrices  1  0  0   0   0 0

0 A 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 J tA−1 J 0

0 0 0 0 0 1

      

where A ∈ Sp(n − 2). The connected center C1 is the subgroup of the matrices        

c1 0 0 0 0 0

0 c2 I 0 0 0 0

0 0 c3 0 0 0

0 0 0 c−1 3

0 0 0 0

0 0

c−1 2 I 0

0 0 0 0 0 c−1 1

       

where c1 , c2 , c3 ∈ GL(1). The Lie algebra n1 ⊂ so(2n) of H1u is the subalgebra of the matrices skew symmetric with respect to the skew diagonal       

0 M1 M2 M4 M6 0

0 0 M3 M5 M7 t − M6 J

0 0 0 0 −J tM5 −M4

0 0 0 0 −J tM3 −M2

0 0 0 0 0 t − M1 J

0 0 0 0 0 0

      

where M7 is a square (n − 2)-matrix skew symmetric with respect to the skew diagonal. Let H be a wonderful subgroup with the given spherical system, H = H u L, H ⊂ H1 and L ⊂ L1 . Analogously to the odd case we have that dim H1 − dim H = rank Σ − rank Σ/∆′ + rank S p /∆′ = n. Moreover, dim C = 1. The subgroup (L, L) of (L1 , L1 ) can be isomorphic only to Sp(n − 2) and hence it is equal to (L1 , L1 ). In particular, dim H1u − dim H u = n − 2. The (L, L)-module n1 decomposes into three isotypic components: the first one is m1 +m3 +m5 +m6 where these modules are all simple L1 -modules of dimension n−2 (the standard representation Cn−2 or its dual), the second component is a simple L1 -submodule of dimension ((n − 2)(n − 3)/2) − 1 and the third component is a direct sum of three simple L1 -submodules of dimension 1 (the sum of these two last isotypic components is m2 + m4 + m7 ). Notice that if n = 4 the second component is zero. The Lie subalgebra n ⊂ n1 is an L-submodule of codimension n − 2. Since it is a Lie subalgebra, it is the sum of m2 + m4 + m6 + m7 and a submodule m ⊂ m1 + m3 + m5 of codimension n − 2. The subgroup C1 acts as follows, for all c ∈ C1 −1 −1 −1 and for all Mi ∈ mi : c.M1 = c2 c−1 1 M1 , c.M3 = c3 c2 M3 , c.M5 = c3 c2 M5 . −1 −1 −1 −1 ◦ 2 Therefore, C = {c2 c1 = c3 c2 = c3 c2 } = {c3 = 1, c1 = c2 } and m is diagonal in m1 + m3 + m5 , namely defined by an equation µ1 M1 + µ3 M3 + µ5 M5 = 0 with µi 6= 0, hence unique up to conjugation. Therefore L ∼ = CG · C · Sp(n − 2), and Analogously to the odd case H is determined up to conjugation. Existence We can suppose n odd, the even case follows by localization. The subgroup H is spherical. Indeed, consider: 0 A g = 0 J tA−1 J ,

44


where A ∈ GL(n) is the following: I n−1 2

A=

0 J n−1

0

0

1 0

0 I n−1

2

2

!

,

then B gHg −1 is open in G, since b + ghg −1 = so(2n). Moreover, H is wonderful, because H = NG (H). We have dim PSup + rank Σ = dim G/H = (n + 1)(n − 1) and rank ∆ − rank Σ = dim C = 1. Since rank Σ ≤ n we have rank S p ≤ 1, but the case of rank S p = 1 gives rank Σ = n and hence S p = ∅, i.e. a contradiction. Therefore, S p = ∅ and rank Σ = n − 1. The spherical system must be cuspidal, hence Σ contains a root of type a2 and it remains only the case of dc∗ (n). dy(p, p) (p ≥ 4). Uniqueness. We have Σ = S. The set ∆′′ = ∆ \ {δα−p−1 , δα−p , − δα′ , δα−′p } is the minimum parabolic subset of ∆, associated to PS p /∆′′ , where p−1

S p /∆′′ = {α1 , . . . , αp−2 , α′1 , . . . , α′p−2 }. We take ∆′ = {δα+1 , δα+2 , . . . , δα+p−2 , δα+p−1 } ⊂ ∆′′ . Now S p /∆′ = ∅ and Σ(∆′ ) = {αp−1 , αp , α′p−1 , α′p }, and so we have that Σ/∆′ = {α1 + α′p−2 , . . . , αi + α′p−i−1 , . . . , αp−2 + α′1 }. The quotient is a parabolic induction of aa(p − 2, p − 2): qe qe q qe qe e qe qe qe @ @ @qe @qe Our G is SO(2p) × SO(2p). Consider Q = PS−p /∆′′ : we have Q = Qu L(Q) where L(Q) = GL(p−1)×GL(1)×GL(p−1)×GL(1). Also, H1 = H1u L1 , where H1u = Qu , (L1 , L1 ) = SL(p − 1) embedded diagonally in GL(p − 1) × 1 × GL(p − 1) × 1 ⊂ L(Q) and C1 is the connected center of L(Q), and dim C1 = 4. The elements of (L1 , L1 ) have this form:     

A 0 0

0 1 0

0

0

0 0 1 0

 

0 0 0

J tA−1 J

   

A  0  , 0  0

0 1 0

0 0 1

0

0



0 0 0

J tA−1 J

where A ∈ SL(p − 1), while C1 is the subgroup of the couples    

c1 I 0 0 0

0 c2 0 0

0 0 c−1 2 0

0 0 0

 

c−1 1 I

   ,  

c3 I 0 0 0

0 c4 0 0

0 0 c−1 4 0

   

0 0 0 c−1 3 I



  

where c1 , c2 , c3 , c4 ∈ GL(1). The Lie algebra n1 of H1u is the subalgebra of the couples of skew symmetric matrices with respect to the skew diagonal 

0  M1   M2 M3

0 0 0 −J tM2

0 0 0 −J tM1

  0 0  0   ,  M4 0   M5 0 M6

0 0 0 −J tM5

0 0 0 −J tM4

 0 0   0  0

where M1 , M2 , M4 , M5 are row (p − 1)-vectors and M3 , M6 are square (p − 1)matrices skew symmetric with respect to the skew diagonal. As usual we take H = H u L with H ⊂ H1 and L ⊂ L1 . We have that dim H1 − dim H = p+2. Moreover, dim C = 1, hence dim(L1 , L1 )−dim(L, L) ≤ p−1. Table 5 shows that (L, L) can be isomorphic only to SL(p−1), SO(p−1) or Sp(p−1) (if p−1 is even), but from the above inequality we get that (L, L) = (L1 , L1 ) ∼ = SL(p − 1). In particular, dim H1u − dim H u = p − 1. The (L, L)-module n1 decomposes into two isotypic components: the first one is


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m1 + m2 + m4 + m5 where each of these is a simple L1 -submodule of dimension p − 1 and of highest weight λ2 (the dual of the standard representation Cp−1 ), the second component is m3 + m6 where these are simple L1 -submodules of dimension V2 p−1 (p − 1)(p − 2)/2 and of highest weight λ1 (the dual of C ). The Lie algebra n ⊂ n1 of H u is an L-submodule of codimension p − 1. Since it is a Lie subalgebra, it is the sum of m3 + m6 and a submodule m ⊂ m1 + m2 + m4 + m5 of codimension p − 1. The subgroup C1 acts as follows, for all c ∈ C1 and for all Mi ∈ mi : −1 −1 −1 −1 −1 c.M1 = c2 c−1 1 M1 , c.M2 = c2 c1 M2 , c.M4 = c4 c3 M4 , c.M5 = c4 c3 M5 . −1 −1 −1 −1 −1 −1 ◦ Therefore, C = {c2 c1 = c2 c1 = c4 c3 = c4 c3 } = {c1 = c3 , c2 = c4 = 1} and m is diagonal in m1 + m2 + m4 + m5 , namely defined by an equation µ1 M1 + µ2 M2 + µ4 M4 + µ5 M5 = 0 with µi 6= 0, hence unique up to conjugation. Since H contains the center CG of the group G, we have L = CG · GL(p − 1), where GL(p+ 1) is embedded diagonally in L(Q). Moreover, H = NG (H) so it is uniquely determined up to conjugation. Existence. L◦ is isomorphic to GL(p − 1). Now B ′ = BGL(p−1) × GL(1) × − BGL(p−1) × GL(1) is a Borel subgroup of L(Q) that satisfies the hypothesis of the Lemma 4.3. Indeed, B ′ ∩ L◦ = TGL(p−1) has an open orbit in Cp−1 . Hence, H is wonderful, since it is spherical and equal to its normalizer. By Lemma 4.1, dim G/H = 2p2 and dim C = 1. Since dim B u = 2p2 − 2p, we have S p = ∅ and rank Σ = 2p = rank S. The spherical system is hence cuspidal and Σ does not contain roots of type am for m ≥ 2, nor roots of type a1 × a1 or dm . A root of type a′1 would cause the spherical system to be decomposable, since it cannot be nonorthogonal to any root of type a1 , but this is not the case, since H is not a direct product. Therefore, Σ = S and by Lemma 4.2(iv) the spherical system is strongly ∆-connected. Since H is not solvable, by the Proposition 3.2 the spherical system cannot be a comb, hence we have dy(p, p) or dz(4, 4) if p = 4. The latter is excluded by the corresponding uniqueness proof. dy(p, (p − 1)) (p ≥ 4). Uniqueness. The set ∆′′ = ∆ \ {δα−p−1 , δα−′ , δα−′p } is p−1

the minimum parabolic subset of ∆ and S p /∆′′ = {α1 , . . . , αp−2 , α′1 , . . . , α′p−2 }. Set ∆′ = {δα+1 , δα+2 , . . . , δα+p−2 , δα+p−1 } ⊂ ∆′′ . We have S p /∆′ = ∅ and Σ/∆′ = {α1 + α′p−2 , . . . , αi + α′p−i−1 , . . . , αp−2 + α′1 }, the quotient spherical system is a parabolic induction of aa(p − 2, p − 2). Let us pose G = SL(p) × SO(2p). We have L(Q) = GL(p−1)×GL(1)×GL(p−1)×GL(1). Here (L1 , L1 ) = SL(p−1) embedded diagonally in GL(p − 1) × 1 × GL(p − 1) × 1 ⊂ L(Q) and dim C1 = 3. We have that dim H1 − dim H = p + 1 and dim C = 1, hence dim(L1 , L1 ) − dim(L, L) ≤ p − 1. As for dy(p, p), the subgroup (L, L) can be isomorphic only to SL(p − 1) and hence it is equal to (L1 , L1 ). In particular, dim H1u − dim H u = p − 1. The (L, L)-module n1 decomposes into two isotypic components: the first one is m1 +m2 +m3 (three simple L1 -submodules of dimension p−1), the second component m4 is a simple L1 -submodule of dimension (p − 1)(p − 2)/2. The Lie algebra n ⊂ n1 of H u is then m4 + m where m ⊂ m1 + m2 + m3 has codimension p − 1. As usual, C is determined and m is diagonal in m1 + m2 + m3 . And L is determined too, since H ⊃ CG . We have H = NG (H), hence determined up to conjugation. dy(p, (p − 2)) (p ≥ 4). Uniqueness. The set ∆′′ = ∆ \ {δα−′ , δα−′p } is the p−1

minimum parabolic subset of ∆ and S p /∆′′ = {α1 , . . . , αp−2 , α′1 , . . . , α′p−2 }. We take ∆′ = {δα+1 , δα+2 , . . . , δα+p−2 , δα+′ } ⊂ ∆′′ , so S p /∆′ = ∅ and Σ/∆′ = {α1 + 1 α′p−2 , . . . , αi + α′p−i−1 , . . . , αp−2 + α′1 }. The quotient spherical system is a

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parabolic induction of aa(p − 2, p − 2). Let us pose G = SL(p − 1) × SO(2p). We have L(Q) = SL(p − 1) × GL(p − 1) × GL(1). Here (L1 , L1 ) = SL(p − 1) embedded diagonally in SL(p − 1) × GL(p − 1) × 1 ⊂ L(Q) and dim C1 = 2. We have that dim H1 − dim H = p and dim C = 1, hence dim(L1 , L1 ) − dim(L, L) ≤ p − 1; so (L, L) = (L1 , L1 ) and dim H1u − dim H u = p − 1. The (L, L)-module n1 decomposes into two isotypic components: the first one is m1 + m2 where these are simple L1 submodules of dimension p − 1, the second one m3 is a simple L1 -submodule of dimension (p − 1)(p − 2)/2. The Lie subalgebra n is m3 + m where m ⊂ m1 + m2 has codimension p − 1. Hence, C is determined and m is diagonal in m1 + m2 . Finally, H = NG (H) and it is uniquely determined up to conjugation. dy(7). Uniqueness. Take ∆′′ = {δα+1 , δα−1 , δα+2 , δα−2 , δα−3 }: we have S p /∆′′ = {α1 , α2 , α5 , α6 }. If ∆′ = {δα−1 , δα−2 , δα−3 }, then S p /∆′ = ∅ and Σ/∆′ = {α1 + α5 , α2 + α6 }. The quotient spherical system is a parabolic induction of aa(2, 2). Our G is SO(14), so L(Q) = GL(3) × GL(1) × GL(3) and (L1 , L1 ) = SL(3) embedded skew diagonally in GL(3) × 1 × GL(3) ⊂ L(Q), with dim C1 = 3. We have dim H1 −dim H = 5 and dim C = 1, hence dim(L1 , L1 )−dim(L, L) ≤ 3. (L, L) can be isomorphic only to SL(3) and hence it is equal to (L1 , L1 ). In particular, dim H1u − dim H u = 3. The Lie subalgebra n ⊂ n1 is the sum of a codimension 3 submodule m diagonal in m1 + m2 + m3 and the rest of the simple L1 -submodules of n1 ; C is determined, H = NG (H) and it is uniquely determined up to conjugation. Existence. By Lemma 4.3, the subgroup H is spherical and wonderful. By Lemma 4.1 we have S p = ∅ and rank Σ = 7. The spherical system is cuspidal with only roots of type a1 . Therefore, Σ = S, the spherical system is strongly ∆-connected and it is not a comb, so it is de6 (7), dy(7), de7 (7) or de8 (7). We can exclude the other cases by the corresponding uniqueness proofs. dz(4, 4). Uniqueness. Take ∆′′ = {δα+1 , δα+2 , δα−2 }, ∆′ = {δα+1 , δα+2 }, so S p /∆′′ = {α2 , α′2 } and S p /∆′ = ∅, Σ/∆′ = {α2 + α′2 }. The quotient by ∆′ is a parabolic induction of aa(1, 1). G is SO(8) × SO(8), so L(Q) = GL(1) × GL(2) × GL(1) × GL(1) × GL(2) × GL(1). (L1 , L1 ) = SL(2) embedded diagonally in 1 × GL(2) × 1 × 1 × GL(2) × 1 ⊂ L(Q) and dim C1 = 6. Here dim H1 − dim H = 7 and dim C = 1, hence dim(L1 , L1 ) − dim(L, L) ≤ 2. Thus (L, L) can be isomorphic only to SL(2) and hence it is equal to (L1 , L1 ). In particular, dim H1u − dim H u = 2. Here n1 has two isotypic components: the first one is the sum of eight simple L1 -submodules of dimension 2 and the second component is composed by six summands of dimension 1. n ⊂ n1 is the sum of m diagonal in m1 + m2 + m3 + m4 + m5 + m6 (Table 3) having codimension 2, and the rest of the simple L1 -submodules. C is analogously determined. H = NG (H) and uniquely determined up to conjugation. Existence. By Lemma 4.3, H is spherical and wonderful. By Lemma 4.1, dim PSup + rank Σ = 32 and rank ∆ − rank Σ = 1, thus S p is empty and rank Σ = 8. The spherical system is hence cuspidal, without roots of type am for m ≥ 2, nor a1 × a1 nor dm . Being irreducible, there are also no roots of type a′1 . So Σ = S and the spherical system is strongly ∆-connected. It cannot be a comb, thus we have dy(4, 4) or dz(4, 4). The former is excluded by the corresponding uniqueness proof. dz(4, 3). Uniqueness. Take ∆′′ = {δα+1 , δα+2 , δα−2 } and ∆′ = {δα+1 , δα+2 }, so p S /∆′ = ∅ and Σ/∆′ = {α2 + α′2 }. The quotient by ∆′ is a parabolic induction of aa(1, 1). G is SL(4) × SO(8) and (L1 , L1 ) = SL(2) embedded diagonally in L(Q). We have dim C1 = 5, dim H1 − dim H = 6 and dim C = 1, hence dim(L1 , L1 ) − dim(L, L) ≤ 2. As usual, (L, L) = (L1 , L1 ) so dim H1u − dim H u = 2.


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The Lie subalgebra n ⊂ n1 is the sum of a codimension 2 submodule m diagonal in the direct sum of five simple L1 -submodules of dimension 2, plus the rest of the simple L1 -submodules. C is determined, H = NG (H) and it is uniquely determined up to conjugation. dz(4, 2). Uniqueness. Take ∆′′ = {δα+1 , δα+2 , δα−2 }, ∆′ = {δα+1 , δα+2 }. The quotient by ∆′ is a parabolic induction of aa(1, 1). Our G is SL(3) × SO(8) and (L1 , L1 ) ∼ = SL(2); also dim C1 = 4 and dim(L1 , L1 ) − dim(L, L) ≤ 2. Again, (L, L) = (L1 , L1 ) hence dim H1u − dim H u = 2. The Lie subalgebra n ⊂ n1 is the sum of a codimension 2 submodule m diagonal in the direct sum of four simple L1 -submodules of dimension 2 and the rest of the simple L1 -submodules. C is determined, and H = NG (H) so it is uniquely determined up to conjugation. dz(4, 1). Uniqueness. By renaming the simple roots as usually, ∆′′ = {δα+1 , δα−1 , + δα′ } and ∆′ = {δα+1 , δα+′ }. The Lie subalgebra n ⊂ n1 is the sum of a codimension 2 2 2 submodule m diagonal in the direct sum of three simple L1 -submodules of dimension 2, plus the rest of the simple L1 -submodules. C is determined and H = NG (H), so it is uniquely determined up to conjugation. de6 (7). Uniqueness. Take ∆′′ = {δα+1 , δα+2 , δα−2 } and ∆′ = {δα+1 , δα+2 }, so S p /∆′′ = {α2 , α5 }, S p /∆′ = ∅ and Σ/∆′ = {α2 + α5 }. The quotient by ∆′ is a parabolic induction of aa(1, 1). Let G be SO(14). We have L(Q) = GL(1) × GL(2) × GL(1)×GL(2)×GL(1), with (L1 , L1 ) = SL(2) embedded diagonally in 1×GL(2)× 1 × GL(2) × 1 ⊂ L(Q) and dim C1 = 5. We have dim H1 − dim H = 6 and dim C = 1, hence dim(L1 , L1 ) − dim(L, L) ≤ 2. (L, L) can be isomorphic only to SL(2) and hence it is equal to (L1 , L1 ). In particular, dim H1u − dim H u = 2. The Lie subalgebra n ⊂ n1 is the sum of a codimension 2 submodule m diagonal in m1 +m2 +m3 +m4 +m5 , plus the rest of the simple L1 -submodules. C is determined, and H = NG (H) so it is uniquely determined up to conjugation. Existence. By Lemmas 4.3 and 4.1, H is spherical and wonderful, with S p = ∅ and rank Σ = 7. The spherical system is cuspidal and with only roots of type a1 . Therefore, Σ = S and the spherical system is strongly ∆-connected. It cannot be a comb, and we have de6 (7), dy(7), de7 (7) or de8 (7). We can exclude the other cases by the corresponding uniqueness proofs. de6 (6). Uniqueness. By renaming the simple roots, ∆′′ = {δα+1 , δα−1 , δα+2 }, ∆′ = + {δα1 , δα+2 }. The quotient by ∆′ is a parabolic induction of aa(1, 1). Here G = SO(12), (L1 , L1 ) ∼ = SL(2); also dim C1 = 4, dim H1 − dim H = 5 and dim C = 1. Since (L, L) = (L1 , L1 ), dim H1u − dim H u = 2. Here n ⊂ n1 and C are determined as before, H = NG (H) and it is uniquely determined up to conjugation. de7 (8). Uniqueness. Take ∆′′ = {δα+1 , δα+2 , δα−2 , δα+3 , δα−3 } and ∆′ = {δα+1 , δα+2 , δα+3 }, so that S p /∆′′ = {α2 , α3 , α5 , α6 } and the quotient by ∆′ is a parabolic induction of aa(2, 2). Our G is SO(16), so L(Q) = GL(1) × GL(3) × GL(3) × GL(1) and (L1 , L1 ) = SL(3) embedded skew diagonally in 1 × GL(3) × GL(3) × 1 ⊂ L(Q). We have dim C1 = 4, dim H1 − dim H = 6 and dim C = 1; since dim(L1 , L1 ) − dim(L, L) ≤ 3 then (L, L) = (L1 , L1 ), therefore dim H1u − dim H u = 3. n ⊂ n1 is the sum of a codimension 3 submodule m ⊂ m1 + m2 + m3 + m4 and the rest of the simple L1 -submodules. Also, m2 = m′2 + m′′2 : now m′2 ∼ = m4 are all = m3 ∼ = m1 ∼ V2 3 ′′ isomorphic to the dual of C , while m2 is isomorphic to the dual of Sym2 C3 . So m is the sum of m′′2 and a submodule diagonal in m1 + m′2 + m3 + m4 . C is determined and H = NG (H), so it is uniquely determined up to conjugation.

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Existence. By Lemmas 4.3 and 4.1, H is spherical and wonderful with Σ = S, the spherical system is strongly ∆-connected and it is not a comb. We have de7 (8) or de8 (8). We can exclude the latter by the corresponding uniqueness proof. de7 (7). Uniqueness. Take ∆′′ = {δα+1 , δα−1 , δα+2 , δα−2 , δα+4 }, ∆′ = {δα+1 , δα+2 , δα+4 }. The quotient by ∆′ is a parabolic induction of aa(2, 2). Let G be SO(14): (L1 , L1 ) ∼ = SL(3) and dim C1 = 3. We have dim H1 − dim H = 3 and dim C = 1, (L, L) = (L1 , L1 ) and dim H1u − dim H u = 3. The Lie subalgebra n ⊂ n1 and C are determined as usual; H = NG (H) and it is uniquely determined up to conjugation. de8 (8). Uniqueness. Take ∆′′ = {δα−1 , δα+2 , δα−2 , δα+3 , δα−3 , δα+4 , δα−4 } and ∆′ = {δα−1 , + δα2 , δα+3 , δα−4 }, so S p /∆′′ = {α2 , α3 , α4 , α6 , α7 , α8 } and the quotient by ∆′ is a parabolic induction of aa(3, 3). Here G = SO(16), L(Q) ∼ = GL(1)× GL(4)× SO(6), (L1 , L1 ) ∼ = SL(4) and dim C1 = 2. We have dim H1 − dim H = 5 and dim C = 1, hence dim(L1 , L1 ) − dim(L, L) ≤ 4. (L, L) = (L1 , L1 ) thus dim H1u − dim H u = 4. The Lie subalgebra n ⊂ n1 is the sum of a codimension 4 submodule m ⊂ m1 + m2 V2 4 and the rest of the simple L1 -submodules. Now m2 ∼ C ⊗ C4 = m′2 + = m′′2 where m′2 ∼ = C4 (the standard representation), while m′′2 is simple with = m1 ∼ ′′ dim m2 = 20. So m is the sum of m′′2 and a submodule diagonal in m1 + m′2 . C is determined, H = NG (H) and it is uniquely determined up to conjugation. Existence. By Lemmas 4.3 and 4.1, H is spherical and wonderful with Σ = S, the spherical system is strongly ∆-connected and it is not a comb. We have de7 (8) or de8 (8). We can exclude the former by the corresponding uniqueness proof. de8 (7). Uniqueness. We take ∆′′ = {δα+1 , δα−1 , δα+2 , δα−2 , δα+3 , δα−3 , δα+7 }, ∆′ = + {δα1 , δα+2 , δα−3 , δα+7 }. The quotient by ∆′ is a parabolic induction of aa(3, 3). Let G be SO(14): (L1 , L1 ) ∼ = SL(4), dim C1 = 1, dim H1 − dim H = 4 and dim C = 1. (L, L) = (L1 , L1 ) thus dim H1u − dim H u = 4. Here n1 = m1 + m2 + m3 where these are simple L1 modules, of dimensions 4, 6, 20; n is obtained from n1 by removing m1 and C = C1 . Therefore H is uniquely determined up to conjugation. Remaining cases. For the following cases with more than one strongly ∆-connected component we use the same arguments as above. As in type A, the following lemma is useful for the existence parts. We maintain the above notations. Lemma 4.4 ([L3]). If Σ′ ⊂ Σ/∆′ gives an isolated strongly ∆-connected component of the spherical system (S p , Σ, A)/∆′ such that its localization in supp(Σ′ ) is isomorphic to aa(p + q + p) or aa(q), then: - if q = 1, the corresponding spherical root α of type a1 belongs to Σ and ∆(α) ∩ ∆(β) = ∅ for all β ∈ S, - if q > 1, the localization of the spherical system (S p , Σ, A)/∆′ in supp(Σ/∆′ ) is associated to L1 with (L1 , L1 ) ∼ = SL(p+1)×SL(p+ = K ′ ×K ′′ , where K ′ ∼ ′ q), and if the factor of K isomorphic to SL(p + q) acts trivially on n1 /n, the corresponding spherical root γ of type aq belongs to Σ. aa(1) + ds(n), n ≥ 4. Uniqueness. Choose ∆′′ = {δα+1 , δα−1 , δα2 } and ∆′ = so that S p /∆′′ = S \ {αn , αn+1 } and Σ/∆′ = {α1 + . . . + αn }. The quotient by ∆′ is a parabolic induction of aa(n). Let G be SO(2n + 2). In this case we have P ( Q, since Q = N (H1u ) is the parabolic subgroup containing B − associated to S p /∆′′ ∪ {αn }, while P = P−S p /∆′′ . We have L(P ) ∼ = GL(n) × GL(1), and (L1 , L1 ) ∼ = SL(n) with dim C1 = 2. Also, we have dim H1 − dim H = n and dim C = 2. By Table 5 the subgroup (L, L) can be isomorphic only to SL(n), {δα+1 , δα2 },


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SO(n) or Sp(n) (if n is even). If (L, L) ∼ = SO(n), respectively Sp(n), we have that H is included in a parabolic induction of SO(n), respectively Sp(n). In both cases we get a contradiction, since there are no distinguished subsets of ∆ giving rise to the corresponding quotients. Therefore (L, L) = (L1 , L1 ) and dim H1u − dim H u = n. The Lie algebra n1 decomposes into two simple L1 -submodules m1 + m2 of dimensions n and n(n − 1)/2. The Lie subalgebra n is equal to m2 and C = C1 . Existence. The subgroup H is spherical and equal to NG (H), hence wonderful. We have dim PSup + rank Σ = dim B u − (n− 2)(n− 3)/2 + 3 and rank ∆− rank Σ = 2. Hence, S p 6= ∅ and it can be proven that there are no root of type dm and that there is at least one root of type am . Now it is possible to see that S p is equal to {α3 , . . . , αn−1 } or {α2 , . . . , αn−2 } with at least a root of type an−1 . One can easily prove that there cannot be a projective colour, by looking at the projective fibration it would be associated to, hence we have aa(1) + ds(n) or aa(n − 1) + (2−comb). The latter can be excluded since it admits a morphism into a rank zero spherical system whose set of simple roots of type p is equal to {α1 , . . . , αn−2 } ( S p /∆′′ . ac∗ (n) + aa(1). Uniqueness. Let n be even. Take ∆′′ = ∆ \ {δα1 , δα−n+1 } and ′ ∆ = {δα2 , . . . , δα2i , . . . , δαn }, so that S p /∆′′ = S \ {α1 , αn+1 } and Σ/∆′ = {α2 + 2α3 +α4 , . . . , α2i +2α2i+1 +α2i+2 , . . . , αn−2 +2αn−1 +αn , αn+1 }. The quotient by ∆′ is a parabolic induction of dc(n). Let G be SO(2n + 2), we have that N (H1u ) is the parabolic subgroup containing B − associated to S \ {α1 } = S p /∆′′ ∪ {αn+1 }. Here (L1 , L1 ) ∼ = SL(n) and dim C1 = 2. We have dim H1 − dim H = n and dim C = 2, hence (L, L) = (L1 , L1 ). The Lie algebra n1 decomposes into two conjugated simple L1 -submodules of dimension n. The Lie subalgebra n is equal to one of them, and H = NG (H) so it is uniquely determined up to conjugation. Let n be odd and greater than 3. Take ∆′′ = ∆\{δαn , δα+n+1 } and ∆′ = {δα2 , . . ., δαn−1 , δα−n+1 }, so that S p /∆′′ = S \ {αn , αn+1 } and Σ/∆′ = {α1 + . . . + αn }. The quotient spherical system is a parabolic induction of aa(n). Let G be SO(2n + 2), we have that N (H1u ) is the parabolic subgroup containing B − associated to S \ {αn } = S p /∆′′ ∪ {αn+1 }; moreover (L1 , L1 ) ∼ = SL(n) and dim C1 = 2. We have dim H1 − dim H = n(n − 1)/2 and dim C = 2, hence (L, L) is equal to (L1 , L1 ). The Lie algebra n1 decomposes into two simple L1 -submodules m1 + m2 of dimension n and n(n − 1)/2, respectively. The Lie subalgebra n is equal to m1 . The subgroup H is equal to its normalizer and uniquely determined up to conjugation. Existence. The subgroup H is wonderful, we have dim PSup +rank Σ = (n−1)(n+ 1) and rank ∆ − rank Σ = 2. Hence, S p = ∅, rank Σ = n − 1 and rank ∆ = n + 1. We have two cases with two strongly ∆-connected components of type ac∗ (n) and aa(1). One of such cases has a projective element and it can be excluded like in the previous case. Let n be equal to 3. We have three different spherical systems of type ac∗ (3) + aa(1). The proof is analogous to the case of n > 3. In this case we have n = n(n − 1)/2 and we have three subgroups up to conjugation: H ′ with n = m1 , H ′′ = σ(H ′ ) and H ′′′ with n = m2 (σ(H ′′′ ) = H ′′′ ). We can conclude analogously to the case of do(p + 2). 2−comb + ac∗ (3). Uniqueness. Here ∆′′ = {δα+1 , δα3 , δα−4 } and ∆′ = {δα+1 , δα3 }, and the quotient by ∆′ is a parabolic induction of aa(3). Our G is SO(10), and (L1 , L1 ) ∼ = SL(3) with dim C1 = 3. We have dim H1 − dim H = 4 and dim C = 2, hence (L, L) = (L1 , L1 ) and dim H1u − dim H u = 3. The (L, L)-module n1 decomposes into three isotypic components, one is the sum of two L1 -submodules

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m1 + m2 of dimension 3 and of highest weight λ1 . The Lie algebra n ⊂ n1 is the sum of a codimension 3 submodule diagonal in m1 + m2 and the other two isotypic components. Therefore C is determined, H is equal to its normalizer and uniquely determined up to conjugation. Existence. The subgroup H is wonderful. We have dim PSup +rank Σ = dim B u +4 and rank ∆ − rank Σ = 2. Hence, S p = ∅ and there are two strongly ∆-connected components. We know that G/H → G/H1 extends to a dominant G-morphism with connected fibers between the two wonderful embeddings: now 2−comb + ac∗ (3) is the unique spherical system satisfying the above requirements and admitting (S p , Σ, A)/∆′ as quotient. ac∗ (n1 ) + d(n2 ), n2 ≥ 2. For n2 = 2 we have ac∗ (n1 ) + aa(1, 1). Uniqueness. Let n1 be odd and n2 be equal to 2. Take ∆′′ = ∆ \ {δα1 , δαn1 } and ∆′ = {δα2 , . . . , δα2i , . . . , δαn1 −1 }; the quotient by ∆′ is a parabolic induction of two factors respectively of type ac(n1 − 2) and aa(1, 1). We have G = SO(2n1 + 4) and L(Q) ∼ = GL(1) × GL(n1 − 1) × SO(4). The universal cover of (L1 , L1 ) is isomorphic to Sp(n1 − 1) × SL(2). The connected center C1 has dimension 2. We have dim H1 − dim H = n, dim C = 1 and dim(L1 , L1 ) − dim(L, L) ≤ n − 1. Also, (L, L) = (L1 , L1 ) and dim H1u − dim H u = n1 − 1. The Lie subalgebra n ⊂ n1 is the sum of a codimension n1 − 1 submodule m ⊂ m1 + m2 and the rest of the simple L1 -submodules. The submodule m2 decomposes into two simple L1 -submodules m2 = m′2 + m′′2 , with m′′2 ∼ = m1 as (L, L)-modules. The submodule m is the sum of m′2 and a submodule diagonal in m1 + m′′2 . Let n1 be odd and n2 be greater than 2. The proof is completely analogous; the quotient by ∆′ is a parabolic induction of ac(n1 − 2) and d(n2 ). We have G = SO(2n1 + 2n2 ) and L(Q) ∼ = GL(1) × GL(n1 − 1) × SO(2n2 ). The subgroup (L1 , L1 ) is isomorphic to Sp(n1 −1)×SO(2n2 −1). As above, dim C1 = 2, dim H1 − dim H = n, dim C = 1, dim(L1 , L1 ) − dim(L, L) ≤ n − 1 and (L, L) = (L1 , L1 ) with dim H1u − dim H u = n1 − 1. The Lie subalgebra n ⊂ n1 is the sum of m ⊂ m1 + m2 and the rest of the simple L1 -submodules. The submodule m2 decomposes into m2 = m′2 + m′′2 , with m′′2 ∼ = m1 as (L, L)-modules. The submodule m is the sum of m′2 and a submodule diagonal in m1 + m′′2 . Let n1 be even. Let us treat together the cases n2 = 2 and n2 > 2. Take ∆′′ = ∆ \ {δαn1 } and ∆′ = {δα1 , . . . , δα2i+1 , . . . , δαn1 −1 }; the quotient by ∆′ is a parabolic induction of two factors respectively of type ac(n1 − 1) and d(n2 ). Here (L1 , L1 ) ∼ = Sp(n1 ) × SO(2n2 − 1) (if n2 = 2 its universal cover is Sp(n1 ) × SL(2)). Again (L, L) = (L1 , L1 ), and here dim H1u − dim H u = n1 . The Lie subalgebra n ⊂ n1 is the sum of a codimension n1 submodule m′ ⊂ m and the complementary L1 -submodule of m. The submodule m decomposes into two simple L1 -submodules m = m′ + m′′ , with m′′ of dimension n1 . Existence. We have dim PSup + rank Σ = dim B u − (n2 − 1)(n2 − 2) + n1 and rank ∆ − rank Σ = 1. If n2 = 2, S p = ∅, rank Σ = n1 and rank ∆ = n1 + 1. The spherical system is cuspidal and there is a spherical root of type aa(1, 1) on the bifurcation. If n2 > 2, analogously there is a spherical root of type d(n2 ) on the bifurcation and S p is determined. ay(p + q + p) + d(n), n ≥ 2. Uniqueness. Here ∆′′ = ∆ \ {δαp+q , δα−2p+q }, and ′ ∆ = {δα+1 , . . . , δα+p } is such that the quotient is a parabolic induction of aa((p−1)+ (q + 1) + (p − 1)) and d(n). Let G be SO(4p + 2q + 2n). In this case N (H1u ) is the parabolic subgroup containing B − associated to S \ {α2p+q } = S p /∆′′ ∪ {αp+q }.


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In this case (L1 , L1 ) ∼ = SL(p + q) × SL(p) × SO(2n − 1) (if n = 2 its universal cover is SL(p + q) × SL(p) × SL(2)) and dim C1 = 2. We have dim H1 − dim H = p + q and dim C = 2. By Table 5 the subgroup (L, L) can be isomorphic only to SL(p + q) × SL(p) × SO(2n − 1), SL(p + q) × SO(p) × SO(2n − 1), SL(p + q) × SO(p) × SO(2n − 1) (for p even), SL(p + q) × SL(2) (for p = n = 2), SL(p + q) × SL(p) × SO(r) × SO(2n − 1 − r), SL(p + q) × SL(p) × Spin(7) (for n = 5), SL(p + q) × SL(p) × G2 (for n = 4) and some intersections of two of them. Any choice with (L, L) 6= (L1 , L1 ) produces a H contained in a parabolic induction, and none of these cases corresponds to any quotient of our spherical system. Therefore, (L, L) = (L1 , L1 ) and dim H1u − dim H u = p + q. The Lie subalgebra n ⊂ n1 is the sum of a codimension p + q submodule m′ ⊂ m and the complementary L1 submodule of m. The submodule m decomposes into two simple L1 -submodules m = m′ + m′′ , with m′′ of dimension p + q. Existence. We have dim PSup + rank Σ = dim B u − (q − 1)(q − 2)/2 − (n − 1)(n − 2) + 2p + 2 and rank ∆ − rank Σ = 2. We know that G/H → G/H1 extends to a dominant G-morphism with connected fibers between the corresponding wonderful embeddings, hence S p ⊂ S p /∆′ . If q > 2, it can be shown that the inclusion is proper, there is exactly one root of type dn on the bifurcation and one root of type aq . The rest is a strongly ∆-connected component of roots of type a1 . We have only three cases: ay(p + q + p) + d(n), if p = 3 az ∼ (3 + q + 3) + d(n) or if p = 2 ay ∗ (2 + q + 2) + d(n). The proof can be concluded thanks to the corresponding uniqueness parts. If q ≤ 2, we know analogously that there are three strongly ∆connected components and the proof ends by noticing that ay(p + q + p) + d(n) is the unique spherical system admitting (S p , Σ, A)/∆′′ as quotient. ay(p, p) + d(n1 ) + d(n2 ), n1 , n2 ≥ 2. Uniqueness. Here ∆′′ = ∆ \ {δα−p , δα−′p }, and ∆′ = {δα+1 , . . . , δα+p } is such that the quotient is a parabolic induction of aa(p − 1, p−1), d(n1 ) and d(n2 ). Let G be SO(2p+2n1 )×SO(2p+2n2 ); we have (L1 , L1 ) ∼ = SL(p)×SO(2n1 −1)×SO(2n2 −1) and dim C1 = 2. We have dim H1 −dim H = p+1, dim C = 1 and dim(L1 , L1 ) − dim(L, L) ≤ p. As usual (L, L) = (L1 , L1 ) and dim H1u − dim H u = p. The Lie subalgebra n ⊂ n1 is the sum of m ⊂ m1 + m2 and the rest of the simple L1 -submodules. The submodules m1 and m2 decompose into m1 = m′1 + m′′1 and m2 = m′2 + m′′2 , with m′′1 ∼ = m′′2 as (L, L)-modules. The ′ ′ submodule m is the sum of m1 + m2 and a submodule diagonal in m′′1 + m′′2 . Existence. We have dim PSup + rank Σ = dim B u − (n1 − 1)(n1 − 2) − (n2 − 1)(n2 − 2) + 2p + 2 and rank ∆ − rank Σ = 1. If n1 = n2 = 2 the unique possibility is rank Σ = 2p + 2 and S p = ∅. Moreover, we can deduce that the spherical system consists of a strongly ∆-connected component with roots of type a1 on a Dynkin diagram of type Ap × Ap plus two roots of type a1 × a1 on the bifurcations. We can easily exclude the case of comb + aa(1, 1) + aa(1, 1). The are only two other possibilities: ay(p, p) + aa(1, 1) + aa(1, 1) and az(3, 3) + aa(1, 1) + aa(1, 1) if p = 3. The latter can be excluded by uniqueness. If n1 > 2 or n2 > 2, we can deduce that S p 6= ∅ and hence that Σ contains at least one root of type dm . Moreover, no root of type am with m ≥ 2 or a′1 can occur. We can have two roots of type dm , or one root of type dm and one root of type a1 × a1 , in both cases both roots on the bifurcations. All the other spherical roots are of type a1 lying in a strongly ∆-connected component. ay(p, p) + d(n), n ≥ 2. Uniqueness. Take ∆′′ = ∆ \ {δα−p , δα−′p } and ∆′ = {δα+1 , . . . , δα+p }. The quotient by ∆′ is a parabolic induction of aa(p − 1, p − 1) and

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d(n). Let G be SL(p + 1) × SO(2p + 2n). The subgroup (L1 , L1 ) is isomorphic to SL(p) × SO(2n − 1) and dim C1 = 2. We have dim H1 − dim H = p + 1, dim C = 1. The subgroup (L, L) is equal to (L1 , L1 ) and dim H1u − dim H u = p. The Lie subalgebra n ⊂ n1 is the sum of m ⊂ m1 + m2 and the rest of the simple L1 -submodules. The submodule m2 decompose into m2 = m′2 + m′′2 , with m1 ∼ = m′′2 ′ as (L, L)-modules. The submodule m is the sum of m2 and a submodule diagonal in m1 + m′′2 . ay(p, (p − 1)) + d(n), n ≥ 2. Uniqueness. Analogously, set ∆′′ = ∆ \ {δα−′p }.

The distinguished subset ∆′ = {δα+′ , . . . , δα+′p } is such that the quotient spherical 1 system is a parabolic induction of two factors respectively of type aa(p − 1, p − 1) and d(n). Let G be SL(p) × SO(2p + 2n). The subgroup (L1 , L1 ) is isomorphic to SL(p)×SO(2n−1) and dim C1 = 1. We have dim H1 −dim H = p, dim C = 1. The subgroup (L, L) is equal to (L1 , L1 ) and dim H1u − dim H u = p. The Lie subalgebra n ⊂ n1 is the sum of m′ ⊂ m and the complementary L1 -submodule of m. The submodule m decomposes into two simple L1 -submodules m = m′ + m′′ , with m′′ of dimension p. ay(p + q + (p − 1)) + d(n), n ≥ 2. Uniqueness. Take ∆′′ = ∆ \ {δαp , δα−2p+q−1 } and ∆′ = {δα+p+q , . . ., δα+2p+q−1 }. The quotient by ∆′ is a parabolic induction of two factors respectively of type aa((p−1)+q +(p−1)) and d(n). Let G be SO(4p+2q − 2 + 2n). The subgroup (L1 , L1 ) is isomorphic to SL(p) × SL(p + q − 1) × SO(2n − 1) and dim C1 = 2. We have dim H1 − dim H = p and dim C = 2. The subgroup (L, L) is equal to (L1 , L1 ). The Lie subalgebra n ⊂ n1 is the sum of m′ ⊂ m and the complementary L1 -submodule of m. The submodule m decomposes into two simple L1 -submodules m = m′ + m′′ , with m′′ of dimension p. Existence. By Lemma 4.4 the root α4 + . . . + αq+3 belongs to Σ. Moreover, S p ⊃ {α5 , . . . , αq+2 } and there is a root of type dn (a1 × a1 if n = 2) on the bifurcation. We obtain two cases: az ∼ (3+q+2)+aa(1, 1) if p = 3 and ay(p+q+(p−1))+aa(1, 1). The former can be excluded by the corresponding uniqueness proof. dy(p + q + (p − 1)), p ≥ 3. Uniqueness. Here ∆′′ = ∆ \ {δαp+q−1 , δα−2p+q−2 , − δα2p+q−1 } and ∆′ = {δα+1 , . . ., δα+p−1 , δαp }; the quotient by ∆′ is a parabolic induction of aa((p − 2) + (q + 1) + (p − 2)). Let G be SO(4p + 2q − 2). In this case N (H1u ) is the parabolic subgroup containing B − associated to S \ {α2p+q−2 , α2p+q−1 } = S p /∆′′ ∪{αp+q−1 }. The subgroup (L1 , L1 ) is isomorphic to SL(p+q−1)×SL(p−1) and dim C1 = 3. We have dim H1 − dim H = p + q and dim C = 2. The subgroup (L, L) can be isomorphic only to SL(p+ q − 1)× SO(p− 1), SL(p+ q − 1)× Sp(p− 1) (for p odd) and SL(p + q − 1) × SL(p − 1). If (L, L) is isomorphic to one of the above proper spherical subgroups, H is contained in a parabolic induction, with connected quotient. From both the cases we get a contradiction. Therefore, the subgroup (L, L) is equal to (L1 , L1 ) and dim H1u − dim H u = p + q − 1. Existence. We have dim PSup + rank Σ = dim B u − (q − 1)(q − 2)/2 + 2p and rank ∆ − rank Σ = 2. We know that H ⊂ H1 and that the quotient G/H → G/H1 extends to a dominant G-morphism with connected fibers. If q > 2, there are no root of type dm and there is exactly one root of type aq . The rest is a strongly ∆connected component of roots of type a1 . We have only three cases: dy(p+q+(p−1)) and, if p = 4, dy ∼ (4 + q + 3) or dz ∼ (4 + q + 3). The proof can be concluded by uniqueness. If q ≤ 2, we know analogously that there are two strongly ∆-connected


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components and the proof ends by noticing that dy(p + q + (p − 1)) is the unique spherical system admitting (S p , Σ, A)/∆′′ as quotient. dy(p, (p − 1)) + d(n), p ≥ 4 and n ≥ 2. Uniqueness. In this case ∆′′ = ∆ \ − {δαp−1 , δα−′ , δα−′p }, and ∆′ = {δα+1 , . . . , δα+p−1 } is such that the quotient spherical p−1

system is a parabolic induction of aa(p − 1, p − 1) and d(2). Let G be SO(2p − 2 + 2n) × SO(2p). The subgroup (L1 , L1 ) is isomorphic to SL(p − 1) × SO(2n − 1) and dim C1 = 3. We have dim H1 − dim H = p + 1, dim C = 1 and dim(L1 , L1 ) − dim(L, L) ≤ p−1. The subgroup (L, L) is equal to (L1 , L1 ) and dim H1u −dim H u = p − 1. Existence. We have dim PSup + rank Σ = dim B u + 2p and rank ∆ − rank Σ = 1. Analogously, we get only the cases dz(4, 3)+d(n) (if p = 4) and dy(p, (p−1))+d(n). The former is excluded by the corresponding uniqueness proof. dy′ (p + q + (p − 2)), p ≥ 4. Uniqueness. Take ∆′′ = ∆ \ {δαp−1 , δα−2p+q−3 , − δα2p+q−2 } and ∆′ = {δα+1 , . . ., δα+p−2 , δαp+q−1 }. The quotient by ∆′ is a parabolic induction of aa((p − 2) + q + (p − 2)). Let G be SO(4p + 2q − 4). In this case N (H1u ) is the parabolic subgroup containing B − associated to S \ {α2p+q−3 , α2p+q−2 } = S p /∆′′ ∪{αp+q−2 }. The subgroup (L1 , L1 ) is isomorphic to SL(p−1)×SL(p+q−2) and dim C1 = 3. We have dim H1 −dim H = p and dim C = 2. The subgroup (L, L) is equal to (L1 , L1 ) and dim H1u − dim H u = p − 1. Existence. By Lemma 4.4 the root αp−1 + . . . + αp+q−1 belongs to Σ. We get two cases: dy ′ (p + q + (p − 2)) and dz ∼ (4 + q + 2) (if p = 4). The latter is excluded by the corresponding uniqueness proof. ay∗ (2 + q + 2) + d(n), n ≥ 2. Uniqueness. Set ∆′′ = {δα+1 , δα+2 , δα−2 , δα3 , δαq+5 } and ∆′ = {δα+1 , δα+2 }. The quotient by ∆′ is a parabolic induction of two factors respectively of type aa(1+q +1) and d(n). Let G be SO(2q +8+2n). The subgroup (L1 , L1 ) is isomorphic to SL(2) × SL(q + 1) × SO(2n − 1) and dim C1 = 3. We have dim H1 − dim H = 3 and dim C = 2. The subgroup (L, L) is equal to (L1 , L1 ). Existence. By Lemma 4.4 the root α3 + . . . + αq+2 belongs to Σ. Moreover, S p ⊇ {α4 , . . . , αq+1 } and there is a root of type dn (a1 × a1 if n = 2) on the bifurcation. We obtain two cases: ay(2 + q + 2) + d(n) and ay ∗ (2 + q + 2) + d(n). The former can be excluded by the corresponding uniqueness proof. dy∗ (3 + q + 3). Uniqueness. Choose ∆′′ = ∆ \ {δα+1 , δαq+3 , δα+q+6 } and ∆′ = {δα−1 , δα−2 , δα−3 }; the quotient by ∆′ is a parabolic induction of aa(2 + q + 2). Let G be SO(2q + 12). The subgroup (L1 , L1 ) is isomorphic to SL(3) × SL(q + 2) and dim C1 = 3. We have dim H1 − dim H = 4 and dim C = 2. The subgroup (L, L) is equal to (L1 , L1 ). Existence. By Lemma 4.4 the root α4 + . . . + αq+3 belongs to Σ. Moreover, S p = {α5 , . . . , αq+2 }. The unique possible spherical system is dy ∗ (3 + q + 3). dy∗ (3 + q + 2). Uniqueness. Choose ∆′′ = ∆ \ {δαq+2 , δα+q+5 }, and ∆′ = − {δα1 , δα−2 , δα−q+4 } is such that the quotient spherical system is a parabolic induction of aa(2 + q + 2). Let G be SO(2q + 10). The subgroup (L1 , L1 ) is isomorphic to SL(3)× SL(q + 2) and dim C1 = 2. We have dim H1 − dim H = 3 and dim C = 2. The subgroup (L, L) is equal to (L1 , L1 ). dy∼ (4 + q + 3). Uniqueness. Choose ∆′′ = {δα+1 , δα−1 , δα+2 , δα−2 , δα−3 , δα4 } and ′ ∆ = {δα−1 , δα−2 , δα−3 }. The quotient by ∆′ is a parabolic induction of aa(2, 2) and aa(q). Let G be SO(2q + 14). In this case N (H1u ) is the parabolic subgroup containing B − associated to S p /∆′′ ∪ {αq+3 }. The subgroup (L1 , L1 ) is isomorphic

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to SL(3) × SL(q) and dim C1 = 4. We have dim H1 − dim H = 5 and dim C = 2. By Table 5 the subgroup (L, L) is equal to (L1 , L1 ) and dim H1u − dim H u = 3. If q ≥ 2, the Lie subalgebra n ⊂ n1 is the sum of a codimension 3 submodule m ⊂ m1 + m2 + m3 and the rest of the simple L1 -submodules. If q = 1, we can substitute the colours δα4 and δαq +3 with δα+4 and δα−4 , respectively. In the decomposition of n1 we get two other submodules, m4 and m5 , isomorphic to m1 , m2 , m3 as (L1 , L1 )-modules. As above we need three of them where to take a codimension two diagonal submodule. If n contains m3 we have that H u contains the unipotent radical of a proper parabolic subgroup and it is associated to a parabolic induction, but this is not the case. The same thing occurs if n contains m4 + m1 or m5 + m2 . Analogously, if n contains m4 + m2 or m1 + m5 , we have that H is equal to a wonderful subgroup associated to the following, e q e e e e e e e q q q q q q e e e e e e @ e @q e but this is not the case; moreover, notice that in ∆(α4 ) there is a projective element and in this case H would be included (with connected quotient) in a wonderful subgroup associated to a parabolic induction of dy(4, 3). Therefore, there are only two other conjugated cases, where m is included in m1 + m2 + m3 or m4 + m5 + m3 . Existence. We have dim PSup + rank Σ = dim B u − (q − 1)(q − 2)/2 + 8 and rank ∆ − rank Σ = 2. By Lemma 4.4 we get that the root α4 + . . . + αq+3 belongs to Σ. Moreover, S p = {α5 , . . . , αq+2 } and we get only three cases: dz ∼ (4 + q + 3), dy(4 + q + 3) and dy ∼ (4 + q + 3). The former two cases can be excluded by the corresponding uniqueness proofs. ay(2, 2) + 2−comb. Uniqueness. Choose ∆′′ = {δα−1 , δα+2 , δα−2 , δα+3 , δα−3 } and ′ ∆ = {δα−1 , δα−2 , δα−3 }. The quotient by ∆′ is a parabolic induction of aa(1 + 2 + 1). Let G be SO(12). The subgroup (L1 , L1 ) is isomorphic to SL(2) × SL(3) and dim C1 = 3. We have dim H1 − dim H = 4, dim C = 2, (L, L) = (L1 , L1 ) and dim H1u − dim H u = 3. Existence. From dim PSup + rank Σ = dim B u + 6 and rank ∆ − rank Σ = 2 we get S p = ∅, Σ = S. We have two strongly ∆-connected components. The spherical system ay(2, 2) + (2−comb) is the unique that admits (S p , Σ, A)/∆′′ as quotient. ay(2, 1) + 2−comb. Uniqueness. We take ∆′′ = ∆ \ {δα+3 , δα+5 } and ∆′ = − {δα1 , δα−2 , δα−4 }; the quotient by the latter is a parabolic induction of aa(1 + 2 + 1). Let G be SO(10). The subgroup (L1 , L1 ) is isomorphic to SL(2) × SL(3) and dim C1 = 2. We have dim H1 − dim H = 3, dim C = 2, (L, L) = (L1 , L1 ) and dim H1u − dim H u = 3. az(3, 3) + d(n1 ) + d(n2 ), n1 , n2 ≥ 2. Uniqueness. Choose ∆′′ = {δα+1 , δα+2 , δα−2 , δα4 , δα′4 } and ∆′ = {δα+1 , δα+2 }, so that the quotient by ∆′ is a parabolic induction of three factors of type aa(1, 1), d(n1 ) and d(n2 ). Let G be SO(6+2n1 )×SO(6+2n2). The subgroup (L1 , L1 ) is isomorphic to SL(2) × SO(2n1 − 1) × SO(2n2 − 1) and dim C1 = 4. The subgroup (L, L) is equal to (L1 , L1 ) and dim H1u − dim H u = 2. Existence. If n1 = n2 = 2, we have dim PSup + rank Σ = dim B u + 8 and the unique possibility is rank Σ = 8 and S p = ∅. Moreover, we can deduce that the spherical system consists of a strongly ∆-connected component with roots of type


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a1 on a Dynkin diagram of type A3 × A3 plus two roots of type a1 × a1 on the bifurcations. There are only two other possibilities: az(3, 3) + aa(1, 1) + aa(1, 1) and ay(3, 3) + aa(1, 1) + aa(1, 1). Analogously, if n1 > 2 or n2 > 2 we can prove that there are two roots of type dn1 and dn2 on the bifurcations and all the other spherical roots are of type a1 lying in a strongly ∆-connected component. az(3, 2) + d(n1 ) + d(n2 ), n1 , n2 ≥ 2. Uniqueness. We can take ∆′′ = {δα+1 , δα−1 , + δα2 , δα3 , δα′4 } and ∆′ = {δα+1 , δα+2 }. az(3, 3) + d(n), n ≥ 2. Uniqueness. Here ∆′′ = {δα+1 , δα+2 , δα−2 , δα′4 } so that S p /∆′′ = {α2 , α′2 , α′4 , α′5 }, and ∆′ = {δα+1 , δα+2 } is such that the quotient spherical system is a parabolic induction of two factors of type aa(1, 1) and d(n). az(3, 2) +1 d(n) n ≥ 2. Uniqueness. Here ∆′′ = {δα+1 , δα−1 , δα+2 , δα′4 } and ∆′ = + {δα1 , δα+2 }. The quotient by ∆′ is a parabolic induction of two factors of type aa(1, 1) and d(n). az(3, 2) +2 d(n), n ≥ 2. Uniqueness. Here ∆′′ = {δα+1 , δα+2 , δα−2 , δα′3 } and ∆′ = + {δα1 , δα+2 }. The quotient by ∆′ is a parabolic induction of two factors of type aa(1, 1) and d(n). az(3, 1) + d(n), n ≥ 2. Uniqueness. Here ∆′′ = {δα+1 , δα−1 , δα+′ , δα′4 } and ∆′ = 2

{δα+1 , δα+′ }. The quotient by ∆′ is a parabolic induction of two factors of type 2 aa(1, 1) and d(n). az∼ (3 + q + 3) + d(n), n ≥ 2. Uniqueness. Here we choose ∆′′ = {δα+1 , δα+2 , − δα2 , δαq+3 , δαq+7 } and ∆′ = {δα+1 , δα+2 }. The quotient by ∆′ is a parabolic induction of three factors of type aa(1, 1), aa(q) and d(n). Let G be SO(2q + 12 + 2n). We have dim C1 = 5, dim H1 − dim H = 5, dim C = 2 and (L, L) = (L1 , L1 ). In particular, dim H1u − dim H u = 2. If q ≥ 2, the Lie subalgebra n ⊂ n1 is the sum of a codimension 2 submodule m ⊂ m1 + m2 + m3 + m4 and the rest of the simple L1 -submodules. If q = 1, in the decomposition of n1 we get two other modules isomorphic to m1 , m2 , m3 and m4 as (L1 , L1 )-modules. As above we need four of them where to take a codimension 2 diagonal submodule. The proof is completely analogous to that of dy ∼ (4 + q + 3). Existence. By Lemma 4.4 we get that the root α4 + . . . + αq+3 of type aq belongs to Σ. Therefore, if n = 2 we have S p = {α5 , . . . , αq+2 } and there are two cases: az(3, 3) + aa(1, 1) and ay(3, 3) + aa(1, 1). The latter can be excluded by the corresponding uniqueness proof. If n > 2 it can be deduced that S p contains properly α5 , . . . , αq+2 and there is a root of type dn on the bifurcation. az∼ (3 + q + 2) + d(n), n ≥ 2. Uniqueness. We can take ∆′′ = {δα+1 , δα−1 , δα+2 , δα3 , δαq+6 } and ∆′ = {δα+1 , δα+2 }. dz(4, 3) + d(n), n ≥ 2. Uniqueness. Here ∆′′ = {δα+1 , δα+2 , δα−2 , δα′4 } and ∆′ = {δα+1 , δα+2 }. The quotient by ∆′ is a parabolic induction of two factors of type aa(1, 1) and d(n). Let G be SO(8) × SO(6 + 2n). The subgroup (L1 , L1 ) is isomorphic to SL(2) × SO(2n − 1) and dim C1 = 5. We have dim H1 − dim H = 6 and dim C = 1, hence dim(L1 , L1 ) − dim(L, L) ≤ 2. The subgroup (L, L) is equal to (L1 , L1 ) and dim H1u − dim H u = 2. Existence. If n = 2, dim PSup + rank Σ = dim B u + 8 and rank ∆ − rank Σ = 1, hence S p = ∅. We have the following cases: dz(4, 3)+aa(1, 1) and dy(4, 3)+aa(1, 1). The latter can be excluded by the corresponding uniqueness proof. If n > 2, S p 6= ∅ and there is only one root of type dn on a bifurcation.

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dz(4, 2) + d(n), n ≥ 2. Uniqueness. Here ∆′′ = {δα+1 , δα+2 , δα−2 , δα′3 } and ∆′ = The quotient by ∆′ is a parabolic induction of two factors of type aa(1, 1) and d(n). dz∼ (4 + q + 3). Uniqueness. Here ∆′′ = {δα+1 , δα+2 , δα−2 , δα4 } and ∆′ = {δα+1 , δα+2 }. The quotient by ∆′ is a parabolic induction of two factors respectively of type aa(1, 1) and aa(q). Let G be SO(2q + 14). The subgroup (L1 , L1 ) is isomorphic to SL(2) × SL(q) and dim C1 = 6. We have dim H1 − dim H = 6 and dim C = 2. The subgroup (L, L) is equal to (L1 , L1 ) and dim H1u − dim H u = 2. Existence. The root α4 + . . . + αq+3 belongs to Σ. Since dim PSup + rank Σ = dim B u + (q − 1)(q − 2)/2 + 8 and rank ∆ − rank Σ = 2, we get S p = {α5 , . . . , αq+2 }. We have the following three cases: dz ∼ (4 + q + 3), dy(4 + q + 3) and dy ∼ (4 + q + 3). dz∼ (4 + q + 2). Uniqueness. Set ∆′′ = {δα+1 , δα−1 , δα+2 , δα3 }. The distinguished subset ∆′ = {δα+1 , δα+2 } is such that the quotient spherical system is a parabolic induction of two factors respectively of type aa(1, 1) and aa(q). dz(3 + q + 2). Uniqueness. Take ∆′′ = {δα+1 , δα+2 , δα−2 , δα3 }. The distinguished subset ∆′ = {δα+1 , δα+2 } is such that the quotient spherical system is a parabolic induction of aa(1 + q + 1). Let G be SO(2q + 10). The subgroup (L1 , L1 ) is isomorphic to SL(2) × SL(q + 1) and dim C1 = 4. We have dim H1 − dim H = 4 and dim C = 2. The subgroup (L, L) is equal to (L1 , L1 ) and dim H1u −dim H u = 2. Existence. The root α3 + . . .+ αq+2 belongs to Σ. We get two cases: dy(3 + q + 2) and dz(3 + q + 2). dz(3 + q + 1). Uniqueness. Take ∆′′ = {δα+1 , δα−1 , δα2 , δα+q+2 }. The distinguished subset ∆′ = {δα+1 , δα+q+2 } is such that the quotient spherical system is a parabolic induction of aa(1 + q + 1). az(3, 2) + 2−comb. Uniqueness. Set ∆′′ = {δα−1 , δα+2 , δα−2 , δα−4 }. The distinguished subset ∆′ = {δα−1 , δα−2 , δα−4 } is such that the quotient spherical system is a parabolic induction of aa(1, 1). Let G be SO(14). The subgroup (L1 , L1 ) is isomorphic to SL(2) and dim C1 = 5. We have dim H1 − dim H = 6, dim C = 2, (L, L) = (L1 , L1 ) and dim H1u − dim H u = 3. Existence. From dim PSup + rank Σ = dim B u + 7 and rank ∆ − rank Σ = 2 we get S p = ∅, Σ = S. We have two strongly ∆-connected components. The proof can be concluded by noticing that az(3, 2) + (2−comb) is the unique spherical system admitting (S p , Σ, A)/∆′′ as quotient. ae6 (6) + d(n), n ≥ 2. Uniqueness. Take ∆′′ = {δα+1 , δα+2 , δα−2 , δα7 }. The distinguished subset ∆′ = {δα+1 , δα+2 } is such that the quotient spherical system is a parabolic induction of two factors of type aa(1, 1) and d(n). Let G be SO(12 + 2n). The subgroup (L1 , L1 ) is isomorphic to SL(2) × SO(2n − 1) and dim C1 = 4. We have dim H1 − dim H = 5 and dim C = 1, hence dim(L1 , L1 ) − dim(L, L) ≤ 2. The subgroup (L, L) is equal to (L1 , L1 ) and dim H1u − dim H u = 2. Existence. If n = 2, we have dim PSup +rank Σ = dim B u +7 and rank ∆−rank Σ = 1, hence S p = ∅. We have the following cases: ae6 (6)+aa(1, 1) and ae7 (6)+aa(1, 1). If n > 2, S p 6= ∅ and there is a root of type dn on the bifurcation. ae6 (5) + d(n), n ≥ 2. Uniqueness. Take ∆′′ = {δα+1 , δα−1 , δα+2 , δα6 }. The distinguished subset ∆′ = {δα+1 , δα+2 } is such that the quotient spherical system is a parabolic induction of two factors of type aa(1, 1) and d(n). ae7 (7) + d(n), n ≥ 2. Uniqueness. Take ∆′′ = {δα+1 , δα+2 , δα−2 , δα+3 , δα−3 , δα8 }. The distinguished subset ∆′ = {δα+1 , δα+2 , δα+3 } is such that the quotient spherical system is a parabolic induction of two factors respectively of type aa(2, 2) and d(n). Let G

{δα+1 , δα+2 }.


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be SO(14 + 2n). The subgroup (L1 , L1 ) is isomorphic to SL(3) × SO(2n − 1) and dim C1 = 3. We have dim H1 − dim H = 5 and dim C = 1, hence dim(L1 , L1 ) − dim(L, L) ≤ 3. The subgroup (L, L) is equal to (L1 , L1 ) and dim H1u − dim H u = 3. Existence. If n = 2, dim PSup + rank Σ = dim B u + 8 and rank ∆ − rank Σ = 1, hence S p = ∅. It remains only the case of ae7 (7) + aa(1, 1). If n > 2, S p 6= ∅ and there is a root of type dn on the bifurcation. ae7 (6) + d(n), n ≥ 2. Uniqueness. Take ∆′′ = {δα+1 , δα−1 , δα+2 , δα−2 , δα+4 , δα7 }. The distinguished subset ∆′ = {δα+1 , δα+2 , δα+4 } is such that the quotient spherical system is a parabolic induction of two factors respectively of type aa(2, 2) and d(n). 3−comb + aa(q). Uniqueness. Set ∆′′ = {δα+1 , δα−1 , δα2 }. The distinguished subset ∆′ = {δα+1 , δα2 } is such that the quotient spherical system is a parabolic induction of aa((q + 1)). Let G be SO(2q + 6). The subgroup (L1 , L1 ) is isomorphic to SL(q + 1) and dim C1 = 3. We have dim H1 − dim H = q + 2 and dim C = 2. The subgroup (L, L) is equal to (L1 , L1 ). Existence. We have dim PSup + rank Σ = dim B u − (q − 1)(q − 2)/2 + 4 and rank ∆ − rank Σ = 2. We know that H ⊂ H1 and that the quotient G/H → G/H1 extends to a dominant G-morphism with connected fibers. If q > 2, there are no root of type dm and there is exactly one root of type aq . The rest is a strongly ∆-connected component of roots of type a1 . If q ≤ 2, we know analogously that there are two strongly ∆-connected components. 3−comb + 2−comb. Uniqueness. Set ∆′′ = {δα−1 , δα+2 , δα−2 }. The distinguished subset ∆′ = {δα−1 , δα+2 } is such that the quotient spherical system is a parabolic induction of aa(2). Let G be SO(10). The subgroup (L1 , L1 ) is isomorphic to SL(2) and dim C1 = 4. We have dim H1 − dim H = 4 and dim C = 2. The subgroup (L, L) is equal to (L1 , L1 ). Existence. We have that S p = ∅ and there are two strongly ∆-connected components. The spherical system (3−comb) + (2−comb) is the unique that admits (S p , Σ, A)/∆′′ as quotient. ax(1 + p + 1) + d(n), n ≥ 2. Uniqueness. Set ∆′′ = ∆ \ {δαp+1 , δα−p+2 }. The distinguished subset ∆′ = {δα+1 , δα2 } is such that the quotient spherical system is a parabolic induction of two factors respectively aa((p + 1)) and d(n). Let G be SO(2p + 4 + 2n). The subgroup (L1 , L1 ) is isomorphic to SL(p + 1) × SO(2n − 1) and dim C1 = 2. We have dim H1 − dim H = p + 1 and dim C = 2. The subgroup (L, L) is equal to (L1 , L1 ). Existence. We have dim PSup +rank Σ = dim B u −(p−1)(p−2)/2−(n−1)(n−2)+4 and rank ∆ − rank Σ = 2. We know that H ⊂ H1 and that the quotient G/H → G/H1 extends to a dominant G-morphism with connected fibers. If q > 2, there is exactly one root of type dn on the bifurcation and one root of type aq . The rest is a strongly ∆-connected component of roots of type a1 . If q ≤ 2, we know analogously that there are three strongly ∆-connected components. af (4) + d(n), n ≥ 2. Uniqueness. Set ∆′′ = {δα+1 , δα+2 , δα−2 , δα5 }. The distinguished subset ∆′ = {δα+1 , δα+2 } is such that the quotient spherical system is a parabolic induction of two factors respectively aa(2) and d(n). Let G be SO(8 + 2n). The subgroup (L1 , L1 ) is isomorphic to SL(2) × SO(2n − 1) and dim C1 = 3. We have dim H1 − dim H = 3, dim C = 2 and dim(L1 , L1 ) − dim(L, L) ≤ 2. The subgroup (L, L) is equal to (L1 , L1 ). Existence. We have dim PSup + rank Σ = dim B u − (n − 1)(n − 2) + 5 and rank ∆ − rank Σ = 2. We know that H ⊂ H1 and that the quotient G/H → G/H1 extends

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to a dominant G-morphism with connected fibers. There is exactly one root of type dn on the bifurcation and two strongly ∆-connected components of roots of type a1 . 5. Normality of the Demazure embedding In this last section we show that the proof of D. Luna ([L4]) of the conjecture of M. Brion ([B3]) on the normality of the Demazure embedding holds also in type A D. Let us report some notations from [L4]. Let g be a semisimple complex Lie algebra, and let h be a Lie subalgebra of g equal to its normalizer. Moreover, G denotes the connected component of the identity in Aut(g) and H the stabilizer of h in G. If m = dim h, Grm (g) denotes the grassmannian of m-dimensional subspaces of g. The Demazure embedding of G/H is the orbit closure G.h in Grm (g). Rigid wonderful varieties. A wonderful G-variety is said to be rigid if it is an embedding of G/H where H = NG (H). In case of an adjoint semisimple group G of type A D, the rigidity of a wonderful variety can be characterized, as for type A, using the classification: Proposition 5.1. Let G be an adjoint semisimple group of type A D, a wonderful p G-variety is rigid if and only if the set AX of its spherical system (SX , ΣX , AX ) does not contain two elements giving the same functional on ΞX . Proof. If NG (H) and H are different then they are the stabilizers of two points in the open G-orbits of two different wonderful G-varieties X ′ and X associated to different spherical systems. The normalizer NG (H) acts on the set of colours of X, and this action is nontrivial because otherwise the spherical system of X ′ would be equal to the one of p p X. This comes from the fact that SX ′ = SX and AX ′ = AX , in type A and D, implies ΣX ′ = ΣX . But now the action of NG (H) can only exchange couples of colours X moved by the same set of simple (spherical) roots and giving the same functional on ΞX . Vice versa, the spherical systems with two elements of A giving the same functional can be easily treated. They are parabolic inductions of spherical systems with at least one irreducible component equal to one of the following three spherical systems: aa(p + 1 + p), D1(i), dc(n) for n even. Each one of them is associated to a nonrigid wonderful variety, embedding of G/H that is respectively equal to SL(2p+2)/S(GL(p+1)×GL(p+1)), SO(2n)/(GL(1)×SO(2n−2)), SO(2n)/GL(n). Therefore, every spherical system with two elements of A with the same functional is associated to a nonrigid wonderful variety. Normality. Theorem 5.2. Let g be of type A D, for every spherical Lie subalgebra h equal to its normalizer, the Demazure embedding G.h is wonderful. It is known that under these hypotheses, for g of any type, the normalization of the Demazure embedding is wonderful ([B3], [Kn]). Therefore, for us it is sufficient to prove that the Demazure embedding itself is normal. As in [L4] the matter can be reformulated as follows. Let X be a wonderful G-variety, embedding of G/H, m = dim H. The morphism x 7→ gx , where x ∈


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G/H and gx is the Lie algebra of the stabilizer of x, extends to a G-morphism δX : X → Grm (g) (the Demazure morphism). If X is a rigid wonderful variety, the morphism δX : X → δX (X) is birational and finite ([B3]). If δX (X) is normal, then δX : X → δX (X) is an isomorphism. Let z ∈ X denote the unique point fixed by B − , and let Tz (X)γ be the γeigenspace inside the tangent space of X at z, for γ any spherical root. In [L4] it is shown that the normality of δX (X) can be proven checking that the differential of δX , restricted to Tz (X)γ , is injective for all γ. In [L4] the proof goes on as follows. Let G be an adjoint semisimple group of type A. A wonderful G-variety is said to be critical if it is rigid, it is not a parabolic induction (i.e. supp(Σ) ∪ S p = S) and it is either of rank 1 or of rank 2 with at least one simple spherical root (i.e. Σ ∩ S 6= ∅). (i) Let X be a rigid wonderful G-variety. For every spherical root γ there exists a subset S ′ of simple roots such that supp(γ) ∪ S p ⊂ S ′ and the localization of X in S ′ is critical. (ii) If the theorem holds for the critical varieties then it holds in general. (iii) The theorem holds for the critical varieties. The proof of ii in [L4] remains valid for the case A D. We need only to generalize the proofs of i and iii. Proof of i. If γ ∈ / S then it is sufficient to consider the localization in supp(γ). We have five cases, corresponding to rank one spherical systems, four of which occurring in type A (spherical roots of type am m ≥ 2, a′1 , a1 × a1 , d3 ) plus one corresponding to a spherical root of type dm m ≥ 4. Each of these spherical systems is associated to a rigid wonderful variety. 6 If γ = α ∈ Σ ∩ S then there exists a spherical root γ ′ 6= α such that hρ(δα+ ), γ ′ i = hρ(δα− ), γ ′ i. Therefore, it is sufficient to consider the localization in supp{α, γ ′ }. From the list of rank two prime wonderful varieties ([W]), we have five cases, four of which occurring in type A plus one corresponding to γ ′ of type dm m ≥ 3, hρ(δα+ ), γ ′ i = 0 and hρ(δα− ), γ ′ i = −2, namely the rank two spherical system D4(ii). Each of these spherical systems is associated to a rigid wonderful variety. Proof of iii. The rank one cases are symmetric varieties, hence the theorem follows from [DP]. It remains to deal with the above critical cases of rank 2, Σ = {α, γ ′ }, and prove the injectivity of the differential of δX only on Tz (X)α . In [L4] this is reduced to verify that sα · (γ ′ − hρ(δ), γ ′ iα) 6= (γ ′ − hρ(δ), γ ′ iα), for δ ∈ A(α). Hence, we have only to check the last of the five above rank two spherical systems: Σ = {α = α1 , γ ′ = 2α2 +. . .+2αn−2 +αn−1 +αn }. Here we have γ ′ −hρ(δα+ ), γ ′ iα = γ ′ and γ ′ − hρ(δα− ), γ ′ iα = γ ′ + 2α, while sα · γ ′ = γ ′ − hα∨ , γ ′ iα = γ ′ + 2α and sα · (γ ′ + 2α) = γ ′ .

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, On spherical varieties of rank one (after D. Ahiezer, A. Huckleberry, D. Snow), Group actions and invariant theory (Montreal, PQ, 1988), CMS Conf. Proc., 10, Amer. Math. Soc., Providence, RI, 1989, 31–41. , Vers une g´ en´ eralisation des espaces sym´ etriques, J. Algebra 134 (1990), no. 1, [B3] 115–143. [C] Camus, R., Vari´ et´ es sph´ eriques affines lisses, Ph.D. Thesis, Institut Fourier, Universit´ e J. Fourier, Grenoble, 2001. [DP] De Concini, C., and Procesi, C., Complete symmetric varieties, Invariant theory (Montecatini, 1982), Lecture Notes in Math., 996, Springer, Berlin, 1983, 1–44. [D] Delzant, T., Classification des actions hamiltoniennes compl` etement int´ egrables de rang deux, Ann. Global Anal. Geom. 8 (1990), no. 1, 87–112. [Kn] Knop, F., Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc. 9 (1996), no. 1, 153–174. [Kr] Kr¨ amer, M., Sph¨ arische Untergruppen in kompakten zusammenh¨ angenden Liegruppen, Compositio Math. 38 (1979), no. 2, 129–153. [L1] Luna, D., Toute vari´ et´ e magnifique est sph´ erique, Transform. Groups 1 (1996), no. 3, 249– 258. [L2] , Grosses cellules pour les vari´ et´ es sph´ eriques, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., 9, Cambridge Univ. Press, Cambridge, 1997, 267–280. ´ , Vari´ et´ es sph´ eriques de type A, Inst. Hautes Etudes Sci. Publ. Math. 94 (2001), [L3] 161–226. [L4] , Sur les plongements de Demazure, J. Algebra 258 (2002), 205–215. [LV] Luna, D. and Vust, T., Plongements d’espaces homog` enes, Comment. Math. Helv. 58 (1983), no. 2, 186–245. [M] Mikityuk, I.V., Integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Math. USSR-Sb. 57 (1987), 527–546. [W] Wasserman, B., Wonderful varieties of rank two, Transform. Groups 1 (1996), no. 4, 375–403. [B2]

` La Sapienza, P.le Aldo Moro 2, 00185 Roma, Dipartimento di Matematica, Universita Italy Current address: Fachbereich Mathematik, Universit¨ at Wuppertal, Gauss-Str. 20, 42097 Wuppertal, Germany E-mail address: [email protected] ` La Sapienza, P.le Aldo Moro 2, 00185 Roma, Dipartimento di Matematica, Universita Italy Current address: Institut Fourier, Universit´ e Joseph Fourier, B.P. 74, 38402 Saint-Martin d’H` eres, France E-mail address: [email protected]